MANUAI    OF  ADVANCED  OPTICS 


C.    RIB OR G    MAXX 


ASSISTANT  PROFESSOR  OF  PHYSICS  IN  THE  UNIVERSITY  OF  CHICAGO 


CHICAGO 
SCOTT,  FORESMAX  AXD   COMPAXY 

1902 


. 


COPYRIGHT,    1902,    BY 
SCOTT,    FORESMAN  AND  COMPANY 


BOUT.  O.  LAW  CO.,  PRINTERS  AND  BINDERS,  CHICAGO 


TYPOGRAPHY  BY 
MARSH,  AITKEN  A  CURTIS  COMPANY,  CHICAGO 


INTRODUCTORY  NOTE 

Anyone  who  has  not  used  the  methods  of  measurement  which 
are  based  upon  the  interference  of  light  waves  will  find  it  difficult 
to  appreciate  the  high  degree  of  accuracy  which  can  thereby  be 
attained.  That  these  methods  are  not  more  commonly  used 
seems  to  be  due  in  large  measure  to  the  fact  that  they  have 
hitherto  not  received  adequate  treatment  in  the  texts  which  are 
in  general  use  in  physical  laboratories.  For  this  reason  this 
"Manual  of  "Advanced  Optics,"  in  which  these  methods  are  for 
the  first  time  presented  in  text-book  form  to  students  of  >hysics, 
is  very  timely,  and  should  prove  a  valuable  aid  in  making  these 
very  practical  and  useful  optical  methods  familiar  to  all  who  may 
at  any  time  find  it  necessary  to  make  measurements  of  great  pre- 
cision. It  is  also  hoped  that  the  book  will  serve  to  promote 
interest  in  the  general  study  of  experimental  optics.  Those  who 
desire  to  enter  into  optical  investigation  can  not  get  a  better  foun- 
dation for  future  work  than  by  studying  the  optical  theories  here 
presented,  and  performing  the  experimeuts  described. 

A.    A.    MlCHELSOtf. 

THE  UNIVERSITY  OF  CHICAGO, 
November  20,  1902. 


117487 


PREFACE 

That  the  practical  study  of  optics  has  been  somewhat  crowded 
out  of  physical  laboratories  by  the  demands  of  electricity  is 
attested  by  the  fact  that  there  have  already  been  published  many 
excellent  manuals  of  the  latter  branch  of  science,  whereas  a  prac- 
tical treatise  on  optics  has  not  yet  appeared.  To  be  sure,  the 
theory  of  optics  needs  no  better  treatment  than  it  has  received  at 
the  hands  of  Mascart,  Prude,  Bassett,  Preston,  or  of  the  authors 
of  the  Winkelmann's  Handbuch  der  Physik.  But  from  what  book 
can  the  student  find  out  how  to  determine,  for  example,  that  most 
important  constant  of  the  spectrometer,  the  resolving  power?  Or 
where  can  he  learn  to  study  practically  the  methods  of  using  the 
phenomena  of  interference  for  exact  measurements? 

This  small  manual  is  an  attempt  to  meet  the  needs  of  the  more 
advanced  students  of  optics.  It  has  been  written  primarily  for 
the  use  of  the  author's  classes  in  the  University  of  Chicago,  and  it 
covers  the  work  done  by  them  during  three  months  of  their  senior 
year.  It  is  hoped,  however,  that  it  will  be  found  useful  at  other 
universities  and  will  serve  as  a  stimulus  to  a  more  extensive  study 
of  this  most  fascinating  branch  of  science. 

Every  chapter  begins  with  a  brief  discussion  of  the  theory  of 
the  experiments  which  follow.  In  this  discussion  the  attempt  has 
been  made  to  avoid  as  much  as  possible  the  use  of  mathematics — 
to  present  rather  the  physical  ideas  involved,  and  to  use  those 
ideas  in  building  up  a  concrete  conception  of  the  phenomena  with 
which  we  are  dealing.  This  has  often  resulted  in  a  lack  of  rigor 
of  demonstration,  e.g.,  in  the  case  of  the  grating.  In  all  such 
cases  references  have  been  added  so  that  those  who  wish  the  com- 
plete and  rigorous  demonstration  can  satisfy  themselves  of  the  cor- 

5  • 


tf  PREFACE 

rectness  of  the  conclusions.  This  course  has  been  followed 
because  the  author  believes  that  clear  conceptions  of  fundamental 
ideas  are  absolutely  indispensable  to  the  physicist,  and  that  the 
mathematical  discussion,  though  often  very  elegant  and  conve- 
nient, adds  nothing  essential  to  these  conceptions,  but  tends, 
rather,  if  used  too  freely,  to  cause  one  to  forget  the  real  essence 
of  the  subject. 

It  is  hoped  that  the  descriptions  of  the  manipulations  are  not 
so  detailed  as  to  reduce  the  student  to  a  mere  mechanical  copyist. 
He  should  not  be  allowed  to  forget  that  there  may  be  other 
methods  of  adjustment  and  manipulation  which  may  be  better 
than  those  suggested.  The  numerical  examples  are  all  taken  from 
the  note-books  of  the  students  who  have  taken  the  course. 

The  apparatus  needed  for  the  experiments  is  not  expensive.  A 
spectrometer  with  such  accessories  as  a  grating,  a  pair  of  Fresnel 
mirrors  and  a  bi-prism,  a  pair  of  Mcol  prisms,  and  some  doubly 
refracting  crystals  can  be  used  for  a  large  number  of  the  exercises. 
The  other  necessary  pieces  are  an  interferometer  and  a  polariscope 
or  saccharimeter.  Every  well  equipped  laboratory  should  have  all 
of  these  things,  and  so  the  course  avoids  the  objection  of  requir- 
ing elaborate  and  expensive  apparatus. 

The  author's  own  work  is  so  arranged  that  a  course  of  lectures 
on  optical  theory  accompanies  the  laboratory  work  outlined  in  this 
manual.  The  two  courses  are  independent,  and  of  such  a  nature 
that  either  can  be  taken  without  the  other.  It  is,  however,  very 
advantageous  to  take  both.  For  the  benefit  of  those  who  take 
the  laboratory  work  only,  the  last  two  chapters  on  the  develop- 
ment of  optical  theory  and  the  trend  of  modern  optics  have  been 
added.  Though  such  chapters  are  not  usual  in  a  laboratory 
manual,  it  is  believed  that  they  will  serve  the  purpose  of  adding 
unity  to  tne  course,  by  placing  each  experiment  in  its  proper  con- 
nection with  the  general  body  of  the  subject,  so  that  the  whole 
will  appear  in  good  perspective. 


PREFACE  7 

The  author  does  not  claim  any  great  originality  for  his  work. 
He  has  followed  to  a  large  extent  the  method  of  presentation 
developed  by  Mr.  A.  A.  Michelson  and  used  by  him  in  his  lec- 
tures. He  is  also  greatly  indebted  to  the  papers  of  Lord  Ray- 
leigh,  and  to  the  treatises  mentioned  above,  as  well  as  to  Kayser's 
monumental  work,  Handbuch  der  Spectroscopie.  He  is  also  under 
obligations  to  his  colleagues  at  the  University  of  Chicago.  He 
received  the  course,  somewhat  in  the  shape  in  which  he  has  now 
written  it  out,  from  his  predecessor,  Mr.  S.  W.  Stratton,  and  has 
received  during  the  five  years  in  which  it  has  been  growing  into 
its  present  form  many  suggestions  from  the  other  members  of  the 
Physics  Department.  He  further  wishes  especially  to  thank  Mr. 
F.  B.  Jewett  for  making  the  drawings  for  the  illustrations. 

RYERSON  PHYSICAL  LABORATORY 

The  University  of  Chicago, 

September  15,  1902. 


CONTENTS 

PAGE 

PREFACE   ...       .       .  .        .        .       .       -       .       .       ft 

I.    LIMIT  OF  RESOLUTION 11 

II.    THE  DOUBLE  SLIT 19 

III.  THE  FRESNEL  MIRRORS        .....  .30 

IV.  THE  FRESNEL  Bi- PRISM .       .      44 

V.    THE  MICHELSON  INTERFEROMETER     .  .48 

VI.  THE  VISIBILITY  CURVES          .        .  .       .       .        .70 

VII.  THE  PRISM  SPECTROMETER .83 

VIII.  TOTAL  REFLECTION   .        .        .        .        .     (  .        .        .        .105 

IX.  THE  DIFFRACTION  GRATING         .       .       .       .       .       .        108 

X.  THE  CONCAVE  GRATING   .        ...       .       .  .116 

XI.  POLARIZED  LIGHT 127 

XII.  ROTATION  OF  THE  PLANE  OF  POLARIZATION  .    130 

XIII.  ELLIPTICALLY  POLARIZED  LIGHT 140 

XIV.  THE  REFLECTION  OF  POLARIZED  LIGHT  FROM  HOMOGENEOUS 

TRANSPARENT  SUBSTANCES      .        .  .        .        .        .    150 

XV.    METALLIC  REFLECTION         .       .       .        .  .       .        .        156 

XVI.    THE  SPECTROPHOTOMETER *•       .    159 

XVII.    THE  DEVELOPMENT  OF  OPTICAL  THEORY  ...        165 

XVIII.    THE  TREND  OF  MODERN  OPTICS      .       .  .       .       .       .172 

APPENDIX     ..       ..       .       ..       .       .  .       .       .        183 

INDEX  194 


, 


MANUAL    OF   ADVANCED 
OPTICS 


LIMIT   OF  RESOLUTION 

Theory 

Since  diffraction  phenomena  play  a  very  important  part  in 
all  optical  instruments,  it  will  be  well  to  begin  the  study  of 
practical  optics  with  a  discussion  of  some  of  the  cases  of  diffrac- 
tion which  occur  most  frequently.  The  simplest  method  of 
observing  these  diffraction  phenomena  is  that  devised  byFraun- 


FlGURE  1 

hofer.*  The  observation  is  most  easily  made  by  placing  before 
a  telescope,  focused  for  parallel  rays,  a  screen  containing  the 
opening  whose  diffraction  pattern  is  to  be  studied,  and  directing 
the  telescope  at  a  distant  point  or  line  source.  The  advantage  of 

*Fraunhofer,  Neue  Modification  des  Lichtes,  Denks.  d.  k.  Akad.  d. 
Wiss,  zu  Munchen,  Vol.  VIII.,  p.  1, 1821.  Gesammelte  Werke,  p.  53;  also 
Harper's  Scientific  Memoirs,  II,  Prismatic  and  Diffraction  Spectra,  p. 
13,  1898. 

11 


12  MANUAL   OF   ADVANCED    OPTICS 

this  procedure  is  apparent-  because  only  those  rays  which  are 
parallel  to  each  other  in  front  of  the  screen  come  to  a  focus  at  a 
given  point,  and  therefore  the  measurement  of  the  diffraction 
pattern  reduces  to  a  measurement  of  angle  only. 

The  simplest  case  is  that  of  a  rectangular  opening. 

Let  AB  (Fig.  1)  represent  a  section  of  the  screen  by  a  plane 
perpendicular  to  two  of  the  sides  of  the  rectangular  opening,  and 
ab  the  section  of  the  opening.  Suppose  that  the  source  emits 
monochromatic  light  only,  and  is  a  bright  line  perpendicular  to 
the  plane  of  the  drawing  and  so  placed,  with  reference  to  the 
screen,  that  the  rays  when  they  reach  the  screen  form  a  parallel 
beam  which  falls  normally  upon  the  opening.  L  represents  the 
objective  of  the  telescope,  and  FF'  its  principal  focal  plane. 

The  image  of  the  line  source  formed  in  this  focal  plane  FF' 
will  consist  of  a  central  bright  band  bordered  on  either  side  by 
alternate  dark  and  bright  bands,  these  latter  decreasing  rapidly  in 
intensity  as  they  are  more  remote  from  the  central  bright  band. 

Let  F  represent  the  position  of  the  center  of  the  central 
bright  band,  and  Fr  that  of  the  center  of  the  first  dark  band. 
We  wish  to  find  the  value  of  the  angle  FLF'  =  0  subtended  at  the 
center  of  the  lens  L  by  F  and  F'.  Through  a  draw  db,  perpen- 
dicular to  LF,  and  ad,  perpendicular  to  LF'.  Then  ab  represents 
the  wave  front  of  that  plane  wave  which  is  brought  to  a  focus  at  F. 
All  points  of  this  wave  front  are  in  the  same  phase,  and  hence  a 
bright  band  is  produced  at  F.  Along  ad,  however,  the  various 
points  will  be  in  different  phases,  the  variation  depending  on  the 
angle  which  ad  makes  with  ab.  If  now  bd  =  \  (\  -  wave  length), 
the  phases  of  the  successive  points  along  ad  will  vary  from  A  to  0, 
that  is,  the  phase  at  d  will  be  a  whole  period  ahead  of  that  of  the 
wave  front  ab,  all  parts  of  which  are  in  the  same  phase,  while  at  a 
the  phases  of  the  two  coincide.  Hence  every  point  in  one  half  of 
ad  will  have  a  corresponding  point  in  the  other  half  of  ad  whose 
phase  differs  from  its  own  by  £A.  Therefore,  when  the  vibrations 


LIMIT   OF   RESOLUTION  13 

from  all  the  points  along  ad  are  brought  together  at  the  focus  F', 
they  destroy  each  other  in  pairs,  and  darkness  is  the  result. 

Now  the  sine  of  the  angle  between  the  wave  fronts  ab  and  ad  is 

M 
sin  bad  =  — =•• 

ab 

But,  for  the  first  dark  band,  M  =  A,  and  hence,  placing  ab  =  a 
and  letting  the  angle  bad  =  FLF'  =  0,  we  have 

sin  0  =  -•  (1) 

a 

The  illumination  will  again  reach  a  minimum  when  bd  =  2A, 
oA,  .  .  .  wA,  so  that  the  general  condition  for  a  dark  band  in  the 
diffraction  pattern  formed  in  parallel  light  by  a  rectangular  open- 
ing is 

a  sin  0  =  wX,  (2) 

in  which  m  stands  for  any  whole  number. 
3 

If,  however,  bd  =  —A,  the  phases  of  the  points  along  ad  will 
"Z 

o 

vary  from  0  to  —A,  and  we  may  conceive  the  beam  of  light  to  be 

& 

divided  into  three  sections  of  equal  width  by  planes  perpendicular 
to  ad,  each  section  containing  points  whose  phases  vary  over 
half  a  period.  The  corresponding  points  of  two  of  these  sec- 
tions will  be  in  opposite  phases,  and  will  therefore  be  in  condition 
to  destroy  each  other,  while  the  third  section  will  transmit  light 
to  the  focus  of  the  lens  and  produce  a  maximum  of  illumination 
there. 

Similarly,  when  bd  is  any  odd  multiple  of  half  a  wave,  that  is, 

bd  =  -      —A,  the  phases  of  the  consecutive  points  along  ad  will 
So 

vary  from  0  to  -      — A,  and  we  may  conceive  ad  divided  into 
« 

%m  +  1  equal  parts  by  planes  perpendicular  to  ad.     Then  2m  of 


14 


MANUAL   OF   ADVANCED    OPTICS 


these  parts  are  in  condition  to  destroy  each  other  in  pairs,  leaving 
one  part  to  send  light  to  the  focus  of  the  telescope.     Hence  the 

condition  for  a  maximum  of  illumination  is  bd  =  — - — A,  or, 

£ 

.     .,      2m +  1  . 

a  sm  0   =  — - — A.  (3) 

4i 

It  is  to  be  noted  that,  in  determining  the  positions  of  the 
maxima  and  minima  of  illumination,  it  is  necessary  to  consider  only 
the  width  of  the  opening  and  the  difference  in  phase  of  the  two 
rays  passing  *ts  edges. 

If  the  source  consists  of  a  pair  of  bright  lines  parallel  to  each 
other,  the  diffraction  pattern  in  the  focal  plane  of  the  telescope 
will  be  one  which  results  from  the  superposition  of  the  two  pat- 
terns which  correspond  to  the  two  sources  respectively.  Let 
<f>  (Fig.  2)  denote  the  angle  subtended  by  the  centers  of  the  two 


FIGURE  2 


sources  when  viewed  from  the  center  of  the  lens  L,  and  .Fthe 
position  of  the  center  of  the  central  bright  fringe  due  to  one 
source,  and  G  that  of  the  center  of  the  central  bright  fringe  due 
to  the  other  source.  Then  FLG  =  <£,  and  it  is  evident  that  if 
<f>  >  6  the  central  fringe  at  6r,  due  to  the  second  source,  will  be 
separated  from  that  at  F,  due  to  the  first  source,  by  a  dark  line, 
and  it  will  be  possible  to  distinguish  them  from  each  other.  If 
<£  =  0,  the  two  central  bright  fringes  will  just  touch  each  other, 


LIMIT   OF   RESOLUTION  15 

and  it  may,  or  it  may  not,  be  possible  to  perceive  that  we  are 
observing  a  double  source.  If  <j>  <  0,  the  two  central  bright 
fringes  will  overlap,  and  it  will  be  impossible  to  recognize  that 

the  source  is  double.  Hence  <f>  =  6  =  —  is  called  the  limit  of  reso- 
lution. 

It  is  to  be  noted  that  this  limit  of  resolution  depends  only 
on  the  width  of  the  opening  and  the  wave  length  of  the  light 
used.  Also  that,  excepting  the  central  band,  the  angular  separa- 
tion between  a  maximum  and  its  adjacent  minima  is 

sin  0'  —  sin  0  =  — • 


Experiments 

I.    PROVE  THE  EQUATION  sin  0  =  - 

a 

APPARATUS. — The  experiment  is  most  easily  performed  with 
a  regular  spectrometer. 

ADJUSTMENTS. — Telescope  and  collimator  should  be  focused 
for  parallel  rays  and  their  axes  arranged  in  line  so  that  the  image 
of  the  slit  is  seen  in  the  center  of  the  field  of  view  of  the  tele- 
scope. The  slit  should  be  as  narrow  as  the  intensity  of  the 
source  will  permit.  The  screen  containing  the  rectangular  open- 
ing to  be  studied  should  be  placed  in  front  of  the  objective  of  the 
telescope.  The  diffraction  pattern  will  then  appear  in  the  field 
of  view. 

It  is,  of  course,  not  necessary  to  use  a  spectrometer.  A  tele- 
scope and  a  sufficiently  distant  slit  source  will  give  satisfactory 
results. 

MEASUREMENTS. — The  direct  measurement  of  A.  may  be  avoided 
by  using  a  source  of  monochromatic  light  (sodium  burner,  mer- 
cury or  cadmium  vacuum  tube).  However,  inasmuch  as  all  sources 


16  MANUAL   OF    ADVANCED    OPTICS 

of  monochromatic  light  are  comparatively  faint,  much  better 
results  will  be  obtained  by  using  sunlight  or  the  electric  arc. 
A  light  which  is  sufficiently  monochromatic  for  this  purpose  may 
be  obtained  from  sunlight  by  allowing  the  solar  spectrum  to 
fall  across  the  slit  end  of  the  collimator,  so  that  the  Fraunhofer 
lines  are  parallel  to  the  slit.  The  light  which  passes  through  the 
slit  will  be  sufficiently  monochromatic^  and  at  the  same  time  of 
sufficient  intensity  to  form  very  clear  diffraction  patterns. 

If  the  solar  spectrum  has  been  so  placed  that  one  of  the 
Fraunhofer  lines  nearly  falls  upon  the  slit,  we  may  assume  with 
sufficient  accuracy  for  this  purpose  that  the  wave  length  which 
forms  the  pattern  is  the  same  as  that  of  the  Fraunhofer  line 
which  has  been  placed  near  the  slit.  If  the  slit  has  been  placed 
at  random  in  the"  spectrum  of  the  sun  or  the  electric  arc,  the 
wave  length  may  be  measured  by  introducing  between  the  col- 
limator and  the  telescope  a  small  diffraction  grating  having  a 
known  number  of  lines  to  the  millimeter. 

The  width  of  the  opening  may  be  measured  on  the  dividing 
engine  or  with  a  micrometer  microscope  in  the  usual  way.  In 
order  that  0  may  be  large  enough  to  be  measured  with  accuracy, 
a  should  be  small,  say  0.2  to  0.5  mm. 

If  a  spectrometer  has  been  used  to  perform  the  experiment, 
the  angle  0  may  be  read  directly  from  the  graduated  circle  of 
the  instrument.  If  only  a  telescope  has  been  used,  the  angle  may 
be  measured  by  fastening  a  small  mirror  to  the  telescope  and 
measuring  with  a  telescope  and  scale  the  angle  through  which  this 
mirror  is  turned.  In  case  the  vernier  on  the  graduated  circle  of 
the  spectrometer  does  not  read  to  at  least  5"  it  is  better  to  meas- 
ure 0  in  any  case  with  a  telescope  and  scale.  Since  the  angle 
between  two  successive  dark  bands  or  between  two  successive 
bright  bands  is  also  0  [cf.  equation  (2)  and  (3)],  it  is  well  to 
measure  the  angle  which  corresponds  to  ten  or  more  bands  rather 
than  that  which  corresponds  to  a  single  one. 


LIMIT   OF    RESOLUTION  1? 

EXAMPLE 

Angular  width  of  7  bands 57'  29.7" 

Angular  width  of  1  band  (9) 8'  12.8" 

Wave  length  (A)  (measured  by  grating). .  ,0005866  mm. 

Width  of  the  opening  (a) 0.245  mm. 

-  = .00239 
a 

sin  0=. 00239 


II.    DETERMINATION   OF   THE    LIMIT   OF    RESOLUTION    OF   AH 
OPENING  OF  WIDTH  a. 

APPARATUS. — An  ordinary  telescope  and  a  rather  coarse 
grating  are  all  that  is  needed. 

ADJUSTMENTS. — Set  up  before  a  source  of  monochromatic 
light  the  grating  consisting  of  several  slits  about  1  mm.  wide 
and  1  mm.  apart.  Such  a  grating  is  most  easily  obtained  by  the 
method  devised  by  Fraunhofer.  Two  machine  screws  are  set  in  a 
plate  of  metal  so  as  to  be  parallel  to  each  other.  A  wire  whose 
diameter  is  about  half  the  distance  between  the  threads  of  the 
screw  is  wound  tightly  about  the  two  screws  and  soldered  to 
them.  When  the  wires  are  cut  from  one  side  of  the  pair  of 
screws,  the  remaining  wires  form  a  very  good  grating.  A 
telescope,  before  the  objective  of  which  a  screen  containing  an 
opening  of  width  a  has  been  placed,  should  be  focused  upon  the 
grating. 

MEASUREMENTS. — Place  the  telescope  with  the  screen 
before  its  objective  rather  near  the  grating  and  draw  it  gradually 
away,  being  careful  to  keep  the  edges  of  the  slits  of  the  grating 
in  focus.  Find  the  point  at  which  the  observer  first  ceases  to 
be  able  to  distinguish  that  the  source  at  which  he  is  looking  is 
made  up  of  lines,  i.e.,  the  point  at  which  the  source  appears  to 
be  uniformly  illuminated.  At  this  point  the  limit  of  resolutidn 


18  MANUAL   OF   ADVANCED    OPTICS 

has  been  reached.  If  now  d  is  the  distance  between  the  centers 
of  the  line  sources,  and  D  the  distance  between  the  grating  and 
the  screen  before  the  objective  of  the  telescope,  then,  according 
to  the  theory  given  above, 

.£,*; 

D  ~  a' 

The  distance  D  may  be  measured  with  a  tape,  and  the  other 
quantities  as  in  the  experiment  above. 

This  experiment  is  largely  qualitative.  Different  observers 
will  differ  by  5%  or  more  in  determining  the  exact  point  at 
which  the  grating  ceases  to  be  resolved  by  the  telescope. 

EXAMPLE 

Distance  between  openings  (d)  ..........         0.138  cm. 

Distance  from  source  to  screen  (D)  .......  358  cm. 

Width  of  opening  in  screen  (a)  ..........          0.157  cm. 

Wave  length  (sodium)  (A)  ...............  0000589  cm. 

-     =  .000385 


-  =  .000375  * 
a 

*In  connection  with  the  above  the  student  will  do  well  to  read 
Schwerd,  Die  Beugungserscheinungen  aus  den  Fundamentalgesetzen  der 
Undulationstheorie  analytisch  entwickelt,  Mannheim,  1835.  Verdet,  Legons 
d'optique  physique,  Vol.  1,  p.  309  seq.  Rayleigh,  Collected  Works,  Vol.  1, 
p.  488.  Phil.  Mag.  (5)  Vol.  10,  p.  116,  1880. 


II 

THE  DOUBLE  SLIT 
Theory 

In  the  preceding  chapter  the  diffraction  pattern  produced  by 
a  single  rectangular  opening  was  discussed.  Let  us  now  pass  to 
the  case  of  two  such  openings  or  slits  of  equal  width  and  parallel 
to  each  other.  It  is,  in  the  first  place,  quite  evident  that  each 
one  of  these  slits  will  produce  maxima  and  minima  of  illumina- 
tion in  accordance  with  the  above  conditions,  namely,  equations 
(2)  and  (3).  But  in  addition  to  these,  other  maxima  and  minima 


FIGURE  3 


will  be  formed  because  of  the  action  of  the  rays  from  the 
two  slits  upon  each  other.  The  conditions  which  determine  the 
formation  of  these  latter  fringes  may  be  obtained  by  the  same 
process  employed  above. 

The  other  conditions  remaining  the  same  as  in  Fig.  1,  let  db 
and  cd  (Fig.  3)  represent  the  two  slits,  each  of  width  a.  Call 
the  distance  be  between  the  neighboring  edges  of  the  two  slits  #, 
and  let  a  +  b  =  d. 

Let  .Fbe  the  point  at  which  the  wave  front  ad  is  brought  to  a 

19 


20  MANUAL    OF    ADVANCED    OPTICS 

focus,  and  F'  the  corresponding  point  for  some  other  wave  front, 
as  ED.  Draw  BD  from  B  perpendicular  to  L F'9  the  point  B 
being  determined  by  the  condition  aB  =  %d.  If  now  aD  =  A,  3A, 
.  .  .  (2m  +  1)A,  it  will  readily  be  seen,  since  cc'  =  \aD,  that  the 
points  along  DV  will  be  in  opposite  phase  to  the  corresponding 
points  along  c'd' '.  The  conditions  by  which  the  minima  due  to 
the  interaction  of  the  two  slits  are  determined  will,  therefore,  be 

aD  =  X,  3A,   .   .   .   (2m +1)  A. 
If  we  call  the  angle  between  the  wave  fronts  0,  this  is  equivalent  to 

2dsin  0=  (2m +  1)  A.  (4) 

Similarly,  the  conditions  by  which  the  maxima  are  determined 

are 

aD  -  2mA, 
or 

2d  sin  0'  =  2mA.  (5) 

It  will  be  noted  that  the  angle  between  a  maximum  and  its  adja- 
cent minima  is  sin  0'  —  sin  0  =  — /  These  maxima  and  minima 

2d 

which  are  determined  by  equations  (4)  and  (5)  are  called  those 
of  the  second  order. 

It  has  thus  far  been  assumed  that  the  source  was  a  bright 
line,  that  is,  that  it  was  infinitely  narrow.  As  all  physically 
attainable  sources  have  an  appreciable  width,  they  must  be  looked 
on  as  a  series  of  line  sources  tangent  to  one  another  along  their 
length.  Each  of  these  line  sources  will  produce  its  own  set  of 
bands  in  the  focal  plane  of  the  telescope,  so  that  the  diifraction 
pattern  actually  observed  is  really  made  up  by  the  superposition 
of  a  large  number  of  such  diifraction  bands.  If  the  angular 
width  of  the  source,  when  viewed  from  the  center  of  the  lens  L 
(Fig.  3),  is  equal  to  the  angle  between  the  central  bright  fringe 
and  its  first  adjacent  dark  fringe  as  determined  by  equation  (4), 
the  central  bright  fringe  formed  by  the  light  which  comes  from 
one  edge  of  the  source  will  fall  on  the  first  dark  band  formed  by 


THE    DOUBLE    SLIT  21 

light  from  the  other  edge  of  the  source,  and  all  trace  of  the 
fringes  will  disappear.  Hence  it  is  seen  that  these  interference 
bands  produced  by  two  slits,  —  for  that  is  what  these  maxima  and 
minima  of  the  second  order  really  are,  —  may  be  used  to  measure 
small  angular  magnitudes.  That  they  enable  us  to  measure  wave 
length  also  is  evident  from  equations  (4)  and  (5). 

As  a  clear  understanding  of  these  phenomena  of  interference 
is  desirable  for  what  is  to  follow,  it  will  be  necessary  to  introduce 
here  a  more  detailed  discussion  of  them. 

In  order  that  two  transverse  vibrations,  such  as  the  light 
waves  are,  may  produce  interference,  it  is  essential  that  they  have 
the  same  period.  If  they  so  interfere  as  to  produce  darkness, 
they  must  in  addition  have  the  same  plane  of  vibration  and  the 
same  amplitude.  Since  only  those  rays  which  proceed  from  the 
same  vibrating  particle  can  fulfil  these  requirements,  it  is  the 
fundamental  principle  of  all  interference  phenomena  that  only 
those  rays  which  proceed  from  the  same  point  of  the  source'  can 
interfere  with  each  other.  In  order,  therefore,  to  produce  inter- 
ference, it  is  necessary  to  divide  each  ray  into  two,  to  lead  the 
two  over  paths  of  different  lengths  and  to  reunite  them.  Two 
rays  which  result  from  this  operation,  and  which  are,  therefore, 
capable  of  producing  interference,  are  called  congruent  rays. 

Let  us  consider  two  congruent  rays  which  are  traveling  in  the 
same  direction  along  the  same  straight  line.  If  we  let  the  line  be 
the  axis  of  #,  and  consider  that  the  vibrations  take  place  in  the 
xy  plane,  the  displacement  of  a  particle  at  the  point  x  due  to  one 
of  these  rays  will  be  given  by 

=  A   COB  8r,-~ 


and  the  displacement  of  the  same  particle  due  to  the  other  ray  by 

It      z  + 
ijz  =  Az  cos  '27T  ^—  --- 

in  which  the  symbols  have  the  usual  well-known  significance. 


22  MANUAL   OF   ADVANCED    OPTICS 

The  displacement  of  the  particle  due  to  both  rays  will,  there- 
fore, be 

It       x\  It      x  +  8\ 

y  =  y\  +  y*  =  AI  cos  27r  ( -f  -  ^  )  +  A* cos  %*  \jr  — A/ 

This  becomes  by  expanding  the  last  term 

/  8\  /  t      x  \  8    .          /  t      x  \ 

y  =  I  A!+  Az  cos  2  «•—  j  cos  2ir  i  -=  —  —  I  -f  AZ  sin  2ir  —  sm  2-rr  i  -=  -  —  j. 

If  in  this  equation  we  now  introduce  two  new  constants,  A  and 
D,  determined  by  the  conditions 

8  D 

A!  -F  A2  COS  £TT  —  =  A  COS  2?r  — 
A  A 

AZ  sin  STT  —  =  A  sin  2-n-  — 

A  A 

it  reduces  to 

t       x  +  D 


X 

This  equation  tells  us  that  the  superposition  of  two  simple 
harmonic  vibrations  which  are  traveling  in  the  same  direction, 
results  in  a  simple  harmonic  vibration  which  has  the  same  period 
as  the  two,  but  a  different  amplitude  and  phase. 

If  we  solve  the  equations  which  define  A  and  D  f  or  A,  we 
obtain 

cs 

A2  =  A?  +  A*  +  %AiAz  cos  2?r  —  • 

A 

In  the  special  case  in  which  AI  and  A2  are  equal, "this  becomes 


Hence  we  see  that  if  the  difference  of  path  8  is  a  whole  number 

^ 

of  wave  lengths,  i.e.,  if  —  =  1,  2,  3,  etc.,  A2  will  have  its  maxi- 

A 

mum  value.     But  if  8  is  an  odd  number  of  half  wave  lengths, 

,8        135 
i.e.,  if  -  =   -,  -,  -,   etc.,  A  will  be  zero. 


THE    DOUBLE    SLIT  23 

Let  Ob  (Fig.  4)  represent  the  optical  axis  of  a  telescope,  and 
Si  and  S2  two  slits  which  are  perpendicular  to  the  plane  of  the 
drawing.  0  represents  the  center  of  a  slit  source  parallel  to  the 
two  slits  Si  and  S2,  and  separated  from  their  plane  by  a  distance 
bO  =  d.  Consider  0  as  the  center  of  a  coordinate  system  whose  x 


FIGURE  4 

axis  lies  in  the  plane  of  the  drawing  and  is  perpendicular  to  bO, 
and  whose  y  axis  is  parallel  to  the  slits,  i.e.,  perpendicular  to  the 
page  at  0.  Let  r  be  any  point  of  the  source  on  the  axis  of  x, 
0  the  angle  CSiS2,  and  b  the  distance  between  the  centers  of  the 
slits.  The  difference  of  path  D  of  the  two  rays  S^'  and  CS2r, 
which  come  to  a  focus  in  a  direction  0,  will  be  determined  by 


D=0r-  Sir  =  CSZ  +  S2r  -  /Sir. 


But,  since  Or  =  x, 


Therefore 


But  Szr  +  Sir  =  2d,  nearly,  and  CSt  =  b6,  therefore 


But      is  the  angle  subtended  by  Or  at  the  center  of  the  lens.     If 


24  MANUAL   OF    ADVANCED    OPTICS 

we  call  this  angle  £,  we  shall  have,  as  the  difference  of    phase 
between  the  two  rays, 


in  which  p  represents  the  number  of  waves  in  the  distance  b 
between  the  slits. 

The  intensity  in  the  field  of  view  in  the  direction  0  may  be 
obtained  from  equation  (6).  If  we  denote  by  <£  (£)  the  intensity  of 
the  light  which  comes  from  an  elementary  band  of  the  source 
at  r,  of  angular  width  d£,  we  have,  as  the  intensity  due  to 
this  band  alone, 

Ir  =  2$  (I)  COS>TT  (0  +  i)  d£  =  $  (£)  +  <£  (£)  cos  Ip*  (0  +  £)  dt. 
The  total  intensity  in  the  direction  0  is  the  integral  of  this  expres- 
sion, that  is,  if  we  let 


our  equation  representing  the  intensity  becomes 
/=  P  +  C  cos  2pv0  -  S  sin  2 

The  first  term  of  the  right-hand  side  of  this  equation,  when  taken 
between  the  proper  limits,  represents  the  total  amount  of  light 
emitted  from  the  source.  The  values  of  0  which  correspond  to 
the  maximum  and  minimum  values  of  /  are  determined  by  the 

condition  ^  =  0,  or, 
at) 

C  sin  %pirO  +  8  cos  %pirO  =  0. 

From  this  we  easily  deduce  as  the  equation  for  the  intensity  at 
the  maxima  and  minima, 

I=P  ± 


It  has   been  shown  above   that  as  the  width  of  the  source 
changes,  the  fringes  in  the  field  of  the  telescope  pass  through 


THE    DOUBLE    SLIT  25 

various  stages  of  visibility,  being  sometimes  very  distinct,  some- 
times totally  lost.  We  may  define  the  visibility  of  the  fringes  as 
the  difference  in  the  intensities  of  a  maximum  and  a  minimum 
divided  by  the  sum  of  the  same  intensities.  Thus  if  /j  represent 
the  intensity  of  a  maximum,  and  /8  that  of  a  minimum,  and  V 
the  visibility,  we  may  write 


or,  from  the  above, 

F'-^.  (8) 

If  the  source  is  symmetrical  with  respect  to  the  axis  of  y^  S  -  0, 
and  we  have 

\  V=T  (9) 

If  the  source  is,  as  we  have  supposed  it  above,  a  uniformly 
illuminated  rectangular  opening  or  slit,  <£(£)  =  constant,  and 
hence,  if  we  denote  by  a  the  angular  width  of  the  source  when 
viewed  from  the  center  of  the  lens, 


If  the  linear  width  of  the  source  is  «,  a  =  —  ,  and  we  have 

a 

ba 
sin  TT  —= 


The  fringes  will  disappear  whenever  -  -  is  a  whole  number  m, 

A.U 

that  is,  when 

-  <»> 


MANUAL   OF   ADVANCED    OPTICS 


Since  ^  is  the  angular  width  of  the  source,  and  7-  the  limit  of 

resolution  of  an  opening  of  width  #,  this  condition  is  the  same  as 
the  one  previously  mentioned  (p.  20). 

If  we  have  as  a  source  a  pair  of  uniformly  illuminated  slits 
placed  symmetrically  with  respect  to  0  (Fig.  4),  each  of  angular 
width  a  and  separated  from  each  other  by  an  angular  distance  y, 
the  expression  above  must  be  integrated  twice,  once  between  the 

limits  —  ?•  —  ?r»  —  IT  +  sr'  an(^  again  between  +  ~  —  —•>  +  ^  +  —  •  The 
«/$/$<«  to  .    a        #       » 

result  is 


in  which  p'  is  written  to  show  that  it  differs  from  p. 

Thus,  if  the  linear  distance  between  the  centers  of  the  two 

s* 

slits  be  denoted  by  £,  y  =  -7>  and  therefore  the  fringes  will  dis- 

u 

appear  not  only  in  accordance  with  equation  (11)  but  also  when 


13 

p  y  =  r  2" 


2m  -  1 


'      1S?     n 

Zm  -  I  A 


where  b'(=  p'\]  is  written  to  distinguish  it  from  the  b  in  equation 

(11)  and  m  is  the  order  of  the  disappearance. 

Fig.  5  represents  the  two 
curves  expressed  by  equations 
(10)  and  (12).  The  dotted  curve 
corresponds  to  (12),  and  we  see 
that  from  it  we  infer  the  sepa- 
ration of  the  two  sources.  The 
full  curve  corresponds  to  (10), 
and  from  it  the  width  of  each 
individual  source  may  be  ob- 
FIGURE  5  tained.  It  is  also  to  be  noted 


THE    DOUBLE    SLIT  27 

that  the  full  curve  is  an  envelope  of  the  maxima  of  the  dotted 
curve.  * 

Experiments 

MEASURE  THE  WIDTH  OF  A  HARROW  SLIT 

APPARATUS. — To  get  the  best  results  in  making  measure- 
ments with  the  double  slit  it  is  advisable  to  use  a  large  lens  of 
rather  long  focus.  A  four  or  five  inch  telescope  objective  with  a 
focal  length  of  from  one  to  two  meters  answers  very  well.  The 
lens  should  be  mounted  fifteen  or  twenty  meters  from  the  source 
whose  nature  is  to  be  investigated.  The  double  slit  should  be 
mounted  directly  in  front  of  the  lens.  These  slits  must  either  be 
movable  so  that  their  distance  apart  can  be  varied,  or  they  may 
have  a  fixed  distance  apart  and  the  nature  of  the  source  may  be 
changed.  In  the  former  case  the  slits  must  be  mounted  on  a 
plate  and  fitted  with  a  right  and  left  screw  or  with  some  other 
mechanism  which  allows  them  to  be  moved  symmetrically  with 
respect  to  the  center  of  the  lens,  while  in  the  latter  case  they  may 
be  cut  from  a  card.  Slits  about  1  mm.  wide  and  4  or  5  cm.  apart 
are  very  satisfactory. 

ADJUSTMENTS. — Sunlight  or  some  other  bright  light  is 
allowed  to  pass  through  the  openings  whose  dimensions  are  to  be 
determined  and  to  fall  upon  the  lens  before  which  the  slits  have 
been  placed.  The  image  of  the  source  when  observed  with  the 
eyepiece  is  found  to  consist  of  a  series  of  fringes.  Care  must  be 
taken  to  find  the  interference  fringes,  which  are  narrower  and 
clearer  than  the  diffraction  fringes  and  which  lie  in  the  center 
of  the  entire  pattern. 

MEASUREMENTS. — The  observation  may  be  made  either  by 
separating  the  slits  or  by  varying  the  nature  of  the  source  until 
the  fringes  disappear.  In  both  cases  it  is  necessary  to  measure 

*  For  similar  solutions  for  sources  of  other  shapes  see  Michelson,  Phil. 
Mag.  (5)  Vol.  30,  p.  1,  1890.  Mascart,  Traite  d'Optique,  Vol.  3,  p.  567  seq., 
Paris,  1893. 


28  MANUAL   OF    ADVANCED    OPTICS 

the  distance  b  between  the  slits  and  the  width  a  of  the  source. 
If  the  source  is  double,  the  distance  between  the  two  apertures 
which  compose  it  must  also  be  measured.  The  distance  d  from 
the  double  slit  to  the  source  must  also  be  known.  Since  the  wave 
length  A  of  the  light  used  appears  in  the  equation,  it  is  well  to 
filter  the  light  used  so  that  the  rays  which  reach  the  lens  haie 
approximately  one  wave  length.  If  white  light  is  used  the  disap- 
pearance observed  will  be  that  of  the  fringes  of  greatest  bright- 
ness which  correspond  to  the  wave  length  .00055  mm. 

EX'AMPLES 

1.  A  single  slit  was  set  up  at  a  distance  of  2080  cm.  from  the 
double  slit.     Sunlight  was  used.     The  width  of  the  slit  was  set 
at   random   and  the    two  slits    moved    until    the   fringes  disap- 
peared.     The    following  two    sets   of   observations   were   made. 
Only  the  first  disappearance  of  the  fringes  was  taken. 

Wave  length  A 000055  cm. 

Width  of  slit  a  as  measured  directly 0.049  cm. 

Distance  between  slits  b 2.335  cm. 

Distance  d  to  source 2080  cm. 

^  =  .0000236 
b 

I  =  .0000236. 

2.  The  width  of  the  slit  was  then  changed  so  as  to  be  .074  cm. 
and  the  distance  between  the  double  slits  was  found  to  be  1.545 
cm.     Then,  as  above 

^=.0000356, 
o 

|  =  .0000356. 

3.  Using  sunlight  as  before,  the  two  slits  were  set  at  a  fixed 
distance  of  4.075  cm.  apart.     The  width  of  the  source  was  then 


THE    DOUBLE   SLIT 


29 


varied  and  measured  when  the  fringes  disappeared.  In  this  way 
five  disappearances  were  observed  as  follows,  m  denoting  the 
order  of  the  disappearance  : 

m  =  1, 

w  =  2, 

m  =  3, 


%  = 

.277  mm. 

f-t- 

.277  mm. 

m 

11 

.567 

it 

.283 

t  I 

.858 

« 

.286 

it 

1.150 

it 

.287 

it 

1.412 

it 

.282 

mean  .283  mm. 
The  distance  d  was  in  this  case  as  above.     Therefore 

^  =  .0000135, 
|  =  .0000136. 

4.  A  double  source  was  used  whose  two  components  were 
0.053  cm.  apart,  i.e.  ,  c  =  0.053.  The  distance  d  remained  2080  cm. 
Disappearance  of  the  fringes  was  observed  for  the  following 
values  of  V  : 


1.085  cm. 


"    3.324 
"    5.580 


" 


8.015 


b'  =  2.170  cm. 

2.216 
2.232 
2.290 


mean  2.227  cm. 


=  0.0000255, 


l 


=  0.0000247. 


Ill 

THE   FKESNEL   MIRRORS 
Theory 

In  the  preceding  chapters  the  nature  of  the  measurements 
which  can  be  made  by  observing  the  interference  phenomena 
which  arise  when  the  light  from  a  source  is  viewed  with  a  tele- 
scope, before  the  objective  of  which  one  or  two  slits  have  been 
placed,  has  been  discussed.  This  method  of  producing  interfer- 
ence is  encumbered  with  two  serious  objections.  First,  the  rays 
which  interfere  cross  at  a  rather  large  angle,  which  results  in 
making  the  fringes  very  narrow;  and,  second,  the  amount  of  light 
received  by  the  observer  is  small  because  the  source  must  be  very 
narrow  in  order  to  see  fringes  at  all.  These  two  drawbacks  may 
be  obviated  by  suitable  changes  in  the  form  of  the  apparatus. 

It  is  evident  that,  in  the  formation  of  these  bands,  that  por- 
tion of  the  objective  which  is  covered  plays  no  part.  Hence,  it  is 


FIGURE  6 


possible  to  dispense  with  the  entire  objective  excepting  those 
portions  which  are  immediately  behind  the  slits.  These  portions 
need  not,  for  this  purpose,  be  curved,  but  may  be  replaced 
either  by  prisms  as  in  Fig.  Ga  or  by  mirrors  as  in  Fig.  6J. 

30 


THE   FRESX^L   MIRRORS 


31 


Both  these  forms  enable  us  to  increase  the  width  of  the  fringes  by 
changing  the  inclination  of  the  rays  which  meet  at  F.  In 
broadening  the  fringes  by  these  methods  it  is  necessary  either  to 


make  the  distance  MJF  or  S±F  large,  which  is  objectionable  on 
practical  grounds,  or  to  bring  the  mirrors  or  prisms  closer  together. 
This  latter  procedure  changes  the  telescope  with  which  we  began, 


FIGURE  8 


into  one  of  the  two  well-known  forms  of  interference  apparatus,  the 
Fresnel  mirrors  or  the  bi-prism.  The  mirrors  are  shown  diagram- 
matically  in  Figs.  7  and  8,  the  bi-prism  in  Fig.  9.  It  is  evident 


FIGURE  9 


that  these  two  forms  of  interference  apparatus  are  limited  in 
their  possible  applications  because  the  various  parts  can  not  be 
moved  sufficiently  with  reference  to  each  other. 


MANUAL   OF    ADVANCED    OPTICS 


The  angle  between  the  rays  which  interfere  may  be  made 
smaller  and  the  fringes  thereby  enlarged  by  inserting  at  F  a  plane 
parallel  plate  of  glass,  as  in  Fig.  10.  The  ray  J/i F  is  transmitted 


FIGURE  10 

and  the  ray  M2F  reflected  by  the  plate  F,  so  that  by  a  suitable 
inclination  of  this  plate  the  angle  between  the  rays  may  be  made 
as  small  as  desired,  and  hence  the  fringes  may  be  indefinitely 
enlarged. 

A  further  great  improvement  can  be  made  by  replacing  the 
slit  by  another  piece  of  the  plane  parallel  plate  inserted  at  0,  as 
in  Fig.  11.  With  this  arrangement  the  light  is  not  limited  to  a 


beam  that  has  passed  through  a  slit,  but  may  come  from  any 
broad  source.  The  gain  in  illumination  obtained  by  this  simple 
device  is  enormous.  The  instrument  sketched  in  Fig.  11  is  an 
interferometer.  It  is  capable  of  many  variations  of  form,  aAd  can 
be  used  for  a  large  variety  of  delicate  measurements.  By  thus 


THE    FRESNEL   MIRRORS  33 

converting  the  telescope  into  an  interferometer  all  definition  and 
resolution  are  lost,  but  a  great  gain  in  accuracy  is  the  result.* 

We  will  begin  the  discussion  of  the  various  forms  of  inter- 
ferometer with  the  simpler  cases  of  the  Fresnel  mirrors  and 
bi-prism.f  Let  J/i3/2,  Fig.  12,  represent  the  projection  of 


FIGURE  12 


the  two  mirrors  on  a  plane  perpendicular  to  their  line  of  inter- 
section, and  S  the  projection  of  a  line  source  of  monochromatic 
light,  that  source  being  parallel  to  the  intersection  of  the  mirrors. 
Then  61  and  S»  will  represent  the  two  virtual  images  of  S 
formed  by  J/x  and  J/2  respectively.  Let  AB  represent  a  screen 
parallel  to  the  line  joining  the  virtual  sources  /Si  and  Sz. 

The  illumination  upon  the  screen  AB,  due  to  light  reflected 
by  the  mirrors,  will  be  identical  with  that  due  to  two  separate 
sources,  Si  and  S^  each  of  the  same  intensity  as  S.  Since  P 
is  on  the  perpendicular  erected  at  the  middle  of  the  line  /Si$2,  it  is 
equidistant  from  Si  and  /Si,  and  therefore  each  pair  of  congru- 
ent rays  from  the  source  will  arrive  at  P  in  the  same  phase  and 

*Michelson,  Am.  Jour.  Sci.  (3)  39,  p.  115,  1890. 

f  Fresnel:  Memoire  sur  la  Diffraction  de  la  Lumiere,  §.§63-64.  Oeuvres, 
VoL  I,  p.  329  seq.  Mem.  de  VAcad.,  V.,  p.  414,  1826. 


34  MANUAL   OF   ADVANCED    OPTICS 

produce  a  maximum  of  illumination  there.  It  is  necessary  to  find 
what  will  be  the  illumination  at  any  other  point  P'  on  the  screen. 
This  will  clearly  depend  upon  the  difference  in  length  of  the  paths 
of  the  rays  which  arrive  at  P'  from  Si  and  S2  respectively.  This 
difference  of  path  is  expressed  in  the  notation  of  page  23  by 


but  the  illumination  at  P'  is  a  maximum  when  this  difference  of 
path  is  a  whole  number  of  wave  lengths,  m\.  Hence  the  condi- 
tion for  a  maximum  is 

bx 

mX  =  —i 
d 

that  is,  we  have  a  bright  fringe  at  distances  x  from  the  center  P 
equal  to  -rA,  -T-A,  -T~A,  etc.  The  distance  between  the  successive 
maxima  is  seen  to  be  constant.  If  we  call  it  e  we  have 

e  =  *X.         f  (14) 

We  can  use  the  mirrors  then  to  determine  wave  lengths,  for  d 
and  e  are  easily  measured  directly,  and  if  the  distance  from  the 
source  to  the  intersection  of  the  mirrors  be  called/,  and  the  angle 
between  the  mirrors  a,  it  is  readily  seen  that 

b  =  Zf  sin  a  ; 


hence 


A  =      Ze  sin  a.  (14') 


The  measurements  can  be  further  simplified  by  placing  the  source 
at  an  infinite  distance  by  means  of  a  lens.  Then  /  and  d  become 
infinite  together  and  their  ratio  is  unity.  Under  these  con- 
ditions we  have 

A  =  %e  sin  a.  (15) 


THE   FREStfEL   MIRRORS  35 

In  order  to  obtain  good  results  with  the  mirrors  the  conditions 
assumed  in  the  above  discussion  must  be  accurately  fulfilled.  The 
two  mirrors  must  touch  each  other  along  one  edge,  and  that  edge 
must  be  made  to  coincide  with  the  intersection  of  their  reflecting 
surfaces.*  We  have  already  shown  that  when  the  angular  width  of 
the  source,  viewed  from  the  intersection  of  the  mirrors,  is  equal 
to  the  angular  width  of  a  fringe,  viewed  from  the  same  point,  the 
fringes  disappear.  Therefore,  a  narrow  source  is  necessary  for  the 
production  of  distinct  measurable  fringes.  The  effects  of  diffrac- 
tion on  the  phenomena  have  been  discussed  by  "Weberf  and  by 
Struve.J 

Let  us  now  consider  the  effect  of  introducing  a  transparent 
plate  in  the  path  of  one  of  the  rays.  Let  J/i  and  Jf2  (Fig.  13) 


FlGUBE  13 


represent  the  mirrors,  and  Si  and  JS.2  the  virtual  sources  as  in 
Fig.  12.  Suppose  we  introduce  into  the  path  of  one  of  the  rays 
SiP  a  plane  parallel  plate  Q  of  some  transparent  substance, 
whose  index  of  refraction  for  the  monochromatic  light  used  is  //,, 

*Feussner,  Winkelmann,  Handbuch  der  Physik,  Vol.  II,  Pt.  1,  p.  528. 
t  Weber,  Wied.  Ann.,  Vol.  VIII,  p.  407,  1879. 
JStruve,  Wied.  Ann.,  Vol.  XV,  p.  49,  1882. 


36  MANUAL    OF    ADVANCED    OPTICS 

and  whose  thickness  is  t.     The  change  D  in  the  optical  length  of 
the  path  SiP  produced  by  the  plate  Q  will  be 


The  difference  between  the  optical  paths  of  the  two  rays  at  any 
point  of  the  screen  P'  will  then  no  longer  be  expressed  by 

Sf'-  /SiP'=  ^»  but  by 

SzP'~ftlP'=~-(^-l)t. 
This  difference  of  optical  path  is  zero  when 

i\  fd 

X=(fi.-l)   -y> 

by  which  the  distance  through  which  the  central  fringe  of  the 
system  is  shifted  from  its  former  position  P  by  the  introduction 
of  the  plate  Q  is  determined.  But  at  this  distance  x  we  had, 
before  the  insertion  of  the  plate,  a  fringe  determined  by  the 
equation 

ft* 

^•T 

in  which  p  stands  for  any  number,  whole  or  fractional.  If  in  this 
equation  the  value  of  x  found  above  be  substituted  for  #,  we  see 
that  the  central  fringe  of  the  shifted  system  takes  the  place  of 
that  fringe  of  the  original  system  whose  order  is 


It  has  thus  far  been  assumed  that  the  source  emitted  mono- 
chromatic light  only.  If  this  is  not  the  case,  but  waves  of  various 
lengths  are  sent  out,  each  set  of  waves  will  form  its  own  set  of 
fringes  in  accordance  with  equation  (14).  Since  the  distance 
between  the  fringes  which  correspond  to  each  wave  length  is  pro- 
portional to  that  wave  length,  the  resultant  figure  on  the  screen  will 


THE    FRESXEL    MIRRORS  37 

consist  of  the  superposition  of  a  number  of  systems  of  fringes  of 
unequal  breadth.  If  the  mirrors  are  so  arranged,  as  in  Fig.  12, 
that  the  rays  from  the  two  sources  travel  entirely  in  air  before 
reaching  the  screen,  the  position  of  the  central  fringe,  which  is 
determined  by  the  condition,  S.2P'  —  SiP'  =  0,  will,  since  this 
condition  is  independent  of  the  wave  length,  be  the  same  for  all 
colors.  Hence  all  systems  of  fringes  will  agree  in  having  a  bright 
fringe  at  that  point  P  which  is  determined  by  this  equation.  If 
white  light  be  emitted  from  the  source,  the  central  fringe  will  be 
white,  free  from  all  trace  of  color.  Hence  it  is  called  the 
achromatic  fringe.  Since  the  distance  from  P  of  any  other 
bright  fringe  is  proportional  to  the  corresponding  wave  length, 
the  various  systems  of  fringes  will  correspond  with  each  other  at 
no  other  point.  Hence  the  adjacent  fringes  will  be  colored,  their 
color  depending  on  how  the  various  systems  happen  to  overlap  at 
the  point  considered. 

If  the  symmetry  of  the  optical  paths  be  disturbed  by  the 
introduction  of  a  transparent  plate  into  the  path  of  one  of  the 
rays,  as  in  Fig.  13,  the  central  fringe  of  each  system  will  be 

shifted,   as  we  have  seen,   an  amount   (/u.  —  1)  — ,  which  will  be 

different  for  each  wave  length,  because  /n  varies.  In  this  case 
there  will,  in  general,  be  no  point  at  which  the  congruent  rays  of 
all  wave  lengths  will  arrive  in  the  same  phase.  There  will,  there- 
fore, be  no  absolutely  achromatic  fringe.  Nevertheless,  a  system 
of  colored  fringes  may  be  obtained  whose  central  band  appears 
nearly  achromatic.  The  determining  condition  here  is  not  that 
the  difference  in  the  optical  paths  S.2P'  and  /S^P'  equals  zero,  which 
is  manifestly  impossible  because  of  the  dispersion  of  the  plate  §, 
but  that  the  change  in  phase  at  the  point  P' ,  corresponding  to  a 
change  in  wave  length,  be  a  minimum.  Let  x  be  the  distance 
from  P,  the  position  of  the  achromatic  fringe  when  the  plate  Q 
is  out,  to  the  point  P',  where  the  central  fringe  of  the  colored 


38  MANUAL   OF   ADVANCED    OPTICS 

system  appears  when  the  plate  Q  is  in.     The  optical  difference  of 
path  of  the  two  rays  has  been  shown  to  be 

&? -&!»-—  0—1)*. 

But  — =-  is  the  apparent  or  geometrical  retardation,  which  we  will 

call  D',  and  (//,—  l)t  =  Z>,  which  we  will  denote  now  by /(A).    The 
difference  of  phase  at  P'  for  any  wave  length  will,  therefore,  be 


In  order  that  this  difference  of  phase  be  a  minimum,  its  derivative 
with  respect  to  X  must  be  zero.     Performing  the  operation  we  get 


Now  equation  (16)  tells  us  how  much  the  central  fringe  of  the 
system  corresponding  to  any  wave  length  A.  is  shifted  by  the 
introduction  of  the  plate  g,  that  is,  it  is  displaced  till  it  coincides 
with  a  fringe  of  the  order 


But  the  central  fringe  of  the  colored  system  is  thereby  shifted  till 
it  coincides  with  that  fringe  of  the  system  produced  by  wave 
length  A,  whose  order  is 


Hence  the  center  of  the  colored  system  is  displaced  from  the 
shifted  center  of  the  system  corresponding  to  A  by  the  number 
of  fringes  expressed  by  the  equation 


This  displacement  can  readily  be  calculated  if  we  know  the 
form  of  the  function  of  X  for  the  substance  of  which  the  plate  Q 


THE   FRESNEJ,   MIRRORS  39 

consists.  It  has  been  found  that  that  form  of  function  first 
proposed  by  Cauchy  satisfies  the  experimental  facts  very  well  for 
the  visible  spectrum.  Assuming  this  equation,  we  have 


=  A  +  -^ 

A" 


but 

hence 


hence 


Experiments 
MEASUREMENT  OF  WAVE  LENGTHS  WITH  THE  MIRROR 

APPARATUS. — The  observations  with  the  Fresnel  mirrors 
can  be  made  in  several  ways.  In  every  case  it  is,  of  course, 
necessary  to  have  a  pair  of  mirrors  properly  mounted.  The 
mirrors  which  are  usually  furnished  with  the  optical  bench  are 
entirely  satisfactory.  They  are  mounted  as  follows:  A  brass 
plate,  which  can  be  fastened  upon  one  of  the  uprights  of  the 
optical  bench,  serves  as  a  mounting  for  both  mirrors.  One 
mirror  is  so  fastened  to  this  plate  that  it  rests  upon  three  screws, 
and  can  thus  be  adjusted  so  as  to  be  parallel  to  the  other  mirror, 
which  is  rigidly  fastened  upon  a  slide  and  can  be  moved  by  means 
of  a  micrometer  screw  in  a  direction  at  right  angles  to  its  surface. 

Simple  and  effective  mirrors  can  be  made  according  to 
Quincke*  in  the  following  way :  Select  a  piece  of  best  plate  glass 
about  10  cm.  long,  2.5  cm.  wide,  and  3  mm.  thick.  Cut  it  in 
the  middle  into  two  pieces  each  5 "cm.  long.  Blacken  the  rear 

*  Quincke,  Pogg.  Ann.  132,  p.  41,  1867. 


40  MANUAL   OF   ADVANCED    OPTICS 

surfaces  with  shellac  containing  lamp  black,  in  order  to  destroy 
the  reflection  from  the  rear  surface.  Plane  a  heavy  block  of  wood 
smooth  and  flat,  and  arrange  six  balls  of  soft  wax  of  equal  size  upon 
the  block  in  such  a  way  that  each  of  the  two  pieces  of  plate  glass, 
when  placed  upon  them,  will  be  supported  upon  one  ball  along  the 
edge  where  the  two  touch  and  upon  two  balls  along  the  edge 
which  is  farthest  from  the  edge  of  contact.  Dust  the  surfaces 
carefully  and  lay  upon  them  another  piece  of  carefully  dusted 
plate  glass  about  20  cm.  long,  5  cm.  wide,  and  3  mm.  thick. 
Press  firmly  with  one  finger  upon  the  larger  plate  of  glass  directly 
over  the  line  of  contact  of  the  two  smaller  pieces.  The  upper 
glass  will  bend  enough  to  set  the  lower  pair  of  plates  at  a  small 
angle  with  each  other,  and  will  at  the  same  time  keep  their  edges 
of  contact  together.  The  original  apparatus  used  by  Fresnel  was 
of  this  nature. 

In  case  it  is  desired  to  perform  the  experiment  with  divergent 
light,  i.e.,  according  to  equation  (14'),  it  is  necessary  to  have  in 
addition  to  the  mirrors  a  slit,  a  micrometer  microscope,  and  a 
telescope  and  scale.  If  the  experiment  is  to  be  performed  with 
parallel  light,  i.e.,  according  to  equation  (15),  an  ordinary  spec- 
trometer may  be  used  to  advantage,  the  mirrors  taking  the  place 
of  the  prism.  The  angle  a  can  then  be  measured  directly  upon 
the  graduated  circle  of  the  instrument,  and  the  distance  between 
the  fringes  can  be  determined  by  removing  the  objective  of  the 
telescope,  and  measuring  the  angle  which  the  fringes  subtend, 
and  the  distance  from  the  line  of  contact  of  the  mirrors  to  the 
focal  plane  of  the  eyepiece.  The  description  of  the  experiment, 
as  given  below,  applies  to  the  optical  bench.  Its  adaptation  to 
other  methods  is  left  to  the  student. 

ADJUSTMENTS. — First,  the  centers  of  the  slit,  the  mirrors, 
and  the  micrometer  should  be  brought  into  the  same  horizontal 
plane  by  measuring  their  distance  from  the  table  upon  which  the 
apparatus  rests.  Second,  the  slit  and  the  cross-hairs  of  the 


THE   FRESNEL   MIKRORS  41 

micrometer  should  be  made  vertical.  The  slit  may  be  made 
vertical  with  the  aid  of  a  plumb  line.  The  cross-hairs  must 
then  be  adjusted  so  as  to  be  parallel  to  the  slit  by  forming  by  means 
of  a  lens  of  short  focal  length  an  image  of  the  slit  in  the 
plane  of  the  cross-hairs  and  then  rotating  the  micrometer  about 
a  horizontal  axis.  Third,  the  mirrors  should  be  adjusted  so 
that  their  line  of  intersection  coincides  with  the  edges  which 
are  in  contact.  This  can  be  approximately  accomplished  by 
observing  the  image  of  a  straight  edge  in  the  mirrors  and 
adjusting  until  this  image  is  an  unbroken  line.  The  accurate 
adjustment  is  made  with  the  help  of  the  fringes.  The  mirrors 
are  then  set  in  place  so  as  to  reflect  the  light  from  the  slit  to  the 
micrometer.  The  angle  between  the  slit  and  the  plane  of  the 
mirrors  should  not  be  less  than  10°  in  order  to  avoid  complica- 
tions due  to  diffraction  at  the  edges  of  the  mirrors.  On  looking 
into  the  micrometer,  the  slit  being  illumined  with  monochro- 
matic light,  and  altering  slowly  the  angle  between  the  mirrors,  the 
fringes  will  appear  if  the  preliminary  adjustment  has  been  care- 
fully made.  If  the  fringes  do  not  appear,  the  plate  which  carries 
the  mirrors  should  be  turned  about  a  horizontal  axis  to  bring  'the 
intersection  of  the  mirrors  parallel  to^the  slit,  i.e.,  vertical.  If 
the  fringes  even  then  do  not  appear  it  indicates  that  the  intersec- 
tion of  the  planes  of  the  surfaces  of  the  mirrors  does  not  coincide 
with  their  common  edge.  This  may  be  caused  by  not  having 
adjusted  the  mirrors  so  that  both  are  vertical,  or  by  allowing  one 
to  protrude  in  front  of  the  other.  The  fringes  may  then  be  found 
by  turning  the  mirror  which  is  supported  on  three  screws,  about 
a  horizontal  axis,  or  by  moving  the  other  mirror,  which  can  be 
displaced  in  a  direction  perpendicular  to  its  surface,  or  by  both 
operations.  When  the  fringes  appear  they  will  probably  not  be 
parallel  to  the  cross-hairs  of  the  micrometer.  Their  centers  may 
be  made  parallel  to  the  cross-hairs  by  tilting  the  adjustable  mirror 
about  a  horizontal  axis.  They  may  be  made  parallel  to  the  cross- 


42  MANUAL   OF    ADVANCED    OPTICS 

hairs  throughout  their  entire  length  by  tilting  the  plate  which 
carries  both  mirrors,  about  a  horizontal  axis.  Having  thus 
obtained  straight  monochromatic  fringes  parallel  to  the  cross- 
hairs, the  fringes  in  white  light  are  found  by  displacing  the 
mirror  which  is  movable  in  a  direction  normal  to  its  surface. 

The  collimation  axis  of  the  micrometer  should  then  be  brought 
into  parallelism  with  the  line  from  the  intersection  of  the  mirrors 
to  the  micrometer.  Open  the  slit  rather  wide  and  reflect,  by 
means  of  a  small  mirror,  a  beam  of  light  through  the  lower  half 
of  the  slit.  Turn  the  micrometer  about  a  vertical  axis  until  the 
reflection  of  the  slit  upon  the  front  lens  of  the  micrometer  is  seen 
through  the  upper  half  of  the  slit. 

MEASUREMENTS. — The  distance  between  the  fringes  is  meas- 
ured with  the  micrometer.  It  is  well  to  measure  the  distance 
over  which  the  thread  moves  in  passing  ten  to  twenty  fringes. 

The  distance  between  the  intersection  of  the  mirrors  and  the 
slit,  and  between  that  intersection  and  the  cross-hairs  of  the 
micrometer,  can  be  measured  with  a  large  pair  of  dividers  with  a 
sliding  scale,  or  with  a  fine  tape. 

The  angle  between  the  mirrors  may  be  measured  with  a  tele- 
scope and  scale  in  the  usual  way. 

In  order  to  attain  accuracy  with  the  Fresnel  mirrors  it  is 
necessary  to  use  a  bright  source  of  light,  and  to  make  the  angle 
between  the  mirrors  large  enough  to  allow  the  formation  of 
twenty  or  more  fringes.  As  these  fringes  will  be  narrow  a  rather 
high-power  micrometer  is  necessary  in  order  to  count  them 
accurately 

EXAMPLES 

1.  Using  as  a  source  of  light  a  sodium  burner  the  following 
measurements  were  obtained : 

Distance  between  the  fringes e=  .012  cm. 

Distance  from  the  mirrors  to  the  slit /=  29.5  cm. 

Distance  from  the  mirrors  to  the  micrometer . .  .  li  =  51.1  cm. 


THE    FRESNEL   MIRRORS  43 

Hence d  =f+h  =  80.6  cm. 

Distance  from  mirrors  to  scale D  =    163  cm. 

Deflection,  by  observing  the  image  of  the  scale  in 

first  one  mirror  and  then  the  other a  =  2.16  cm. 

Hence sin  a  =  —  =    .0066*25 

"^=      .00878 
A  = .0000582 

2.  The  mirrors  were  mounted  on  a  large  spectrometer  in  place 
of  the  prism.  Sodium  light  was  used  as  a  source.  The  light 
passed  through  the  collimator,  and  was,  therefore,  parallel  when 
it  fell  on  the  mirrors.  The  following  measurements  were 
obtained : 

Angle  between  the  mirrors,  being  %  the  angle  between  the  two 
images  of  the  slit,  a  =  11'  28". 

The  lens  of  the  telescope  was  then  removed  and  the  angle 
subtended  in  the  focal  plane  of  the  eyepiece  by  28  fringes 
measured.  This  was  found  to  be  18'  25".  The  distance  from 
the  focal  plane  of  the  eyepiece  to  the  intersection  of  the  mirrors, 
which  should  coincide  with  the  axis  of  the  instrument,  was  46.21 
cm.  Hence  the  angular  width  of  a  fringe  was  3d". 46,  and  the 
linear  width  e  =  .008835  cm.  Hence  A  =  2e  sin  a  =  .00005897  cm. 


IV 

THE   FKESNEL   BI-PEISM 
Theory 

"VVe  will  now  pass  to  the  consideration  of  the  Fresnel  bi-prism. 
Attention  has  already  been  called  to  this  form  of  interferometer, 
its  derivation  from  the  telescope  being  sketched  in  Fig.  9.  Since 
measurements  with  the  bi-prism  are  most  easily  made  when  the 
source  is  at  an  infinite  distance,  the  formulae  will  be  developed 
for  this  case  only.  Let  ABC,  Fig.  14,  represent  the  projection 


A  E 


FIGURE  14 


of  the  prisms  upon  a  plane  perpendicular  to  their  refracting 
edges,  and  PP'  a  screen  parallel  to  AC.  Suppose  a  beam  of 
parallel  light  to  fall  normally  upon  the  prism  faces  A  C.  Upon 
leaving  the  prisms  the  beam  will  be  divided  into  two  whose  wave 
fronts  are  represented  by  BD  and  BE.  From  B  draw  BP  per- 
pendicular to  PP''.  The  point  P  being  equidistant  from  the  two 
wave  fronts,  will  evidently  be  a  position  of  maximum  illumination. 
It  is  necessary  to  find  what  will  be  the  illumination  at  some  other 
point  P'  of  the  screen.  Let  PP'=  x  and  BP'=  d.  As  in  the 

44 


THE    FRESNEL   BI-PRISM  45 

case  of  the  mirrors,  since  all  points  along  BD  and  BE  are  in  the 
same  phase,  the  illumination  at  P'  will  be  a  maximum  if 

EP'  -DP'  =  m\. 
But  EP  =  d  sin  EBP',  and  DP'  =  d  sin  DBP'.     Hence. 


EP'-DP'  =  2d  cos  sin  P'BP. 

But   if   we   denote  by  a  the   angle   between    the  .wave   fronts, 
DBE  =  TT  -  a;  and  sin  P'BP  =  -j-    Therefore,  since  a  is  small, 

EP'  -  DP'  =  x  sin  a. 
We,  therefore,  have  a  maximum  of  illumination  when 

A          t>\ 

./•  =  -.  --  —.  -  •>  etc. 
sin  a    sin  a 

The  distance  between  the  successive  maxima,  denoted  by  e,  is, 
therefore, 

A 

€'-—  -  » 

sin  a 
from  which  we  get  for  the  wave  length 

A.  =  e  sin.  a.  (18) 

The  same  relations  between  the  width  of  the  source  and  the  visi- 
bility of  the  fringes  hold  for  the  bi-prism  as  for  the  mirrors. 


Experiments 

MEASUREMENT  OF  WAVE  LENGTHS  WITH  THE  BI-PRISM 
APPARATUS. — The  apparatus   needed  for  this  experiment  is 
the  same  as  that  used  in  the  previous  experiment,  with  the  excep- 
tion of  the  mirrors,  these  being  now  replaced  by  a  bi-prism. 

ADJUSTMENTS. — The  adjustments  of  the  bi-prism  are  much 
simpler  than  those  of  the  mirrors.     The  slit  should  be  parallel  to 


46  MANUAL   OF   ADVANCED    OPTICS 

the  common  base  of  the  prisms,  and  their  common  face  should  be 
perpendicular  to  the  path  of  the  light  from  the  slit  to  them. 
This  latter  adjustment  is  made  by  reflecting  a  beam  of  light 
through  the.  lower  half  of  the  slit  and  revolving  the  bi-prism 
about  a  vertical  axis  until  the  reflected  light  is  seen  through  the 
upper  half  of  the  slit. 

The  measurements  are  most  easily  and  accurately  made  with 
the  help  of  a  spectrometer.  Telescope  and  collimator  are  adjusted 
for  parallel  light  and  arranged  so  that  the  image  of  the  slit  falls 
upon  the  cross-hairs  of  the  telescope.  The  bi-prism  is  then 
placed  upon  the  prism  table  of  the  spectrometer  and  adjusted, 
either  in  the  way  described  above,  by  reflecting  light  through  the 
slit,  or  with  the  help  of  a  Gauss  eyepiece,  so  that  it  is  perpendicu- 
lar to  the  common  collimation  axis  of  the  collimator  and  telescope. 
Upon  removing  the  objective  of  the  telescope  the  fringes  will 
appear  in  the  field  of  view.  If  the  slit  is  not  parallel  to  the  com- 
mon base-  of  the  prisms  the  fringes  will  not  be  clear  and  evenly 
spaced  throughout  their  entire  length.  The  slit  should  then  be 
rotated  about  a  horizontal  axis  until  the  fringes  are  clear  and 
evenly  spaced.  In  this  case,  as  in  the  case  of  the  mirrors,  the 
fringes  are  very  narrow,  so  that  a  rather  high-power  micrometer 
is  necessary  to  measure  them  accurately,  and  a  bright  light  is 
indispensable.  In  both  of  these  experiments  the  solar  spectrum 
across  the  slit  will  be  found  to  give  the  greatest  satisfaction  as  a 
source  of  light. 

MEASUREMENTS. — If  a  spectrometer  is  used  the  only  meas- 
urements needed  are  the  distance  between  the  fringes  and  the 
angle  between  the  beams  behind  the  bi-prism.  To  obtain  the 
former  the  angle  through  which  the  telescope  turns  when  the 
cross-hair  passes  over  a  counted  number  of  fringes,  and  the  distance 
between  the  bi-prism  and  the  focal  plane  of  the  micrometer  are 
measured.  From  these  two  observed  quantities  the  linear  dis- 
tance between  the  fringes  is  at  once  calculated.  The  angle 


THE   FKESNEL   BI-PRISM  47 

between  the  beams  is  measured  by  replacing  the  objective  of  the 
telescope  and  setting  the  cross-hair  on  first  one  and  then  the  other 
of  the  images  of  the  slit ;  the  angle  through  which  the  telescope 
has  turned  is  the  desired  angle.  If  the  graduations  upon  the 
circle  of  the  spectrometer  are  not  sufficiently  fine  to  allow  of 
reading  the  angles  accurately,  a  small  mirror,  such  as  is  used  for 
galvanometers,  should  be  mounted  upon  the  telescope  and  the 
angle  read  in  this  mirror  with  a  telescope  and  scale  in  the  usual 
way. 

EXAMPLE 

The  bi-prism  was  placed  in  the  spectrometer  in  place  of  the 
prism,  and  the  angle  between  the  wave  fronts  measured.  The 
value  was 

a  =  18'  21". 

The  objective  of  the  telescope  was  then  removed  and  the  angle 
subtended  in  the  focal  plane  of  the  eyepiece  by  16  fringes  in 
sodium  light  was  found  to  be  13'  10".  Thus  the  angular  width 
of  one  fringe  was  49 "A.  The  distance  from  the  focal  plane  of 
the  eyepiece  to  the  axis  of  the  instrument  was  46.21  cm. 
Hence  the  linear  width  of  a  fringe  was  e  =  .01107  cm.  Hence 

\  «  .00005904  cm. 


THE    MICHELSON   INTERFEROMETER 


Theory 

"We  will  now  pass  to  the  consideration  of  the  interferometer  as 
shown  in  Fig.  11.  This  form  of  instrument  may  be  further 
simplified  by  making  one  plate  perform  the  functions  both  of 
separating  and  reuniting  the  beam.  To  accomplish  this  it  is  only 
necessary  to  turn  the  two  mirrors  into  the  positions  CD  as  shown 
in  Fig.  15.  This  is  known  as  the  Michelson*  interferometer.  Irs 


0 

FIGURE  15 


its  simplest  and  most  efficient  form  it  consists  merely  of  four 
glass  plates.  Let  A,  B,  C,  D  (Fig.  15)  represent  the  projec- 
tions of  the  four  plates  on  a  plane  perpendicular  to  their  surfaces. 


*  Michelson,  Phil.  Mag.  (5)  13,  p.  236. 

48 


THE   MICHELSOX    INTERFEROMETER  49 

Light  from  a  source  S  falls  upon  the  plate  A  at  an  angle  of  inci- 
dence of  approximately  45°.  Plates  A  and  B  are  polished  on 
both  sides  and  should,  to  obtain  the  hest  results,  be  of  best  optical 
glass,  free  from  all  strains.  They  should,  furthermore,  be  cut 
from  the  same  piece,  so  as  to  insure  their  having  the  same  optical 
thickness,  and  their  surfaces  should  be  as  plane  and  as  nearly 
parallel  to  each  other  as  it  is  possible  to  make  them.  The  rear 
surface  of  A  is  coated  with  a  semi-transparent  film  of  silver  or 
platinum,  which  should,  if  the  clearest  interference  bands  are 
desired,  be  of  such  a  thickness  that  it  reflects  half  the  light  inci- 
dent upon  it  to  Z>,  and  transmits  the  other  half  to  C.  D  and 
C  are  two  plane  mirrors,  coated  on  the  front  surface  with  a 
thick  coat  of  platinum  or  silver,  and  so  adjusted  as  to  reflect  the 
light  incident  upon  them  back  over  nearly  the  same  path.  These 
two  reflected  beams  meet  again  on  the  rear  surface  of  the  plate  A 
in  a  condition  suitable  to  the  production  of  interference  bands. 
The  plate  B  is  inserted  to  make  the  two  paths  optically  identical. 

The  observer  looks  into  the  apparatus  from  0.  He  will  see 
the  plate  D  directly,  and  an  image  of  the  plate  C  reflected  by  the 
plate  A.  This  image  of  C  will  appear  in  the  direction  of  the 
plate  D  as  far  behind  A.  as  the  mirror  C  really  is  in  front  of  it. 
The  instrument  is  thus  seen  to  consist  essentially  of  a  film  of  air 
inclosed  between  the  plate  D  and  the  virtual  image  of  the 
plate  C.  Hence  in  discussing  the  interference  phenomena  pro- 
duced by  this  instrument  it  is  necessary  to  consider  only  this  film 
of  air. 

Let  omn  om.2  (Fig.  16)  represent  the  two  plane  mirrors  0 
and  D  whose  intersection  is  projected  at  o,  and  whose  mutual 
inclination  is  <£.  The  illumination  at  any  point  P,  not  necessarily 
in  the  plane  of  the  figure,  will  depend  on  the  mean  difference  of 
phase  of  all  the  pairs  of  congruent  rays  which  reach  P  after 
reflection  from  the  mirrors. 

If  the  source  of  light  is  sufficiently  broad,  the  illumination 


50 


MANUAL    OF    ADVANCED    OPTICS 


FIGURE  16 

at  P  will  be  independent  of  its  distance,  form,  or  position 
Let  us  suppose  that  it  is  a  plane  luminous  surface  which  coin 
cides  with  om-i.  Then  its  image  in  omt  will  coincide  with 
and  its  image  in  omz  will  be  a  plane  symmetrical  to  om 
with  respect  to  om2.  For  every  point  p  of  the  first  surface 
there  is  a  corresponding  point  p'  of  the  second,  which  is  symmet 
rically  placed,  and  in  the  same  phase  of  vibration.  If  w( 
suppress  now  the  source  of  light  and  the  mirrors,  and  replace 
them  by  the  images,  the  eifect  at  any  point  P  is  unaltered 
Consider  now  a  pair  of  points  p,  p' .  Let  8  be  the  angle  forrne( 
by  the  line  joining  P  and^?  (or  //)  with  the  normal  to  the  surfac( 
om2->  8  and  <£  being  both  supposed  small.  The  difference  of 
optical  path  D  will  be 

D  =  Pp'  -  Pp  =pp'  cos  8 

to  quantities  of  the  second  and  higher  order.  If  we  denote  by  2£J 
the  distance  between  the  images  at  the  point  where  they  are  cut 
by  the  line  Pp,  we  will  have  to  a  close  degree  of  approximation 

D  =  2t  cos  8. 


(19) 


The  difference  of  phase  A  at  P  is  '2-n-  - 

A 


Since  cos  8  is  nearly 

equal  to  unity,  if  %t  is  increased  by  A.  by  gradually  separating  the 
images,  D  is  also  increased  by  A,  and  the  difference  of  phase  A  at  P 
passes  through  a  complete  period. 


THE    MICHELSOX    INTERFEROMETER 


51 


In  order  to  find  the  form  of  the  interference  fringes,  let  cdef, 
c'd'e'f  (Fig.  17)  represent  the  two  images,  and  let  their  intersec- 
tion be  parallel  to  cf,  and  their  inclination  be  2<£.  Let  P  be  the 


FIGURE  17 


)oint  considered;  P'  the  projection  of  P  on  the  surface  cdef\ 
and  PB  the  line  forming  with  PP'  the  angle  8.  Draw  P'D 
parallel  to  cf,  and  P'C  at  right  angles,  and  complete  the  rectangle 
BDP'C.  Let  P'PC  =  i,  and  DPP'=  0.  Let  PP'  =  P;  and  call 
the  distance  between  the  surfaces  at  the  point  P',  2/0.  We  have 
then, 

t  =  t0+  CP'  ism  <f>  =  t0+  P  tan  <f>  tan  i, 
'and 


D  =  2 


tan  <£  tan  t)  cos  8, 


or 


f0  +  P  tan  <£  tan  i 


(20) 


We  see  that  in  general  D  has  all  possible  values;  and  hence 
all  phenomena  of  interference  would  be  obliterated. 

Bat  to  an  eye  placed  at  P  the  interference  fringes  will  be 
visible  under  certain  conditions.  Let  db  (Fig.  18)  represent  the 
pupil  of  the  eye.  Since  it  has  appreciable  size  it  will  receive  light 


52  MANUAL    OF    ADVANCED    OPTICS 

not  only  from  p  and  p'9.  but  also  from  other  points  as  pl  and  pf. 
Consider  the  ray  .pa  to  enter  the  pupil  at  one  end  of  a  diameter, 
and  the  ray  pj>,  parallel  to  pa,  to  enter  it  at  the  other  end  of  the 


FIGURE  18 


same  diameter.  Since  these  rays  are  parallel  they  will  come  to  a 
focus  at  the  same  point  on  the  retina.  The  difference  of  phase  at 
this  point  of  the  congruent  rays  ap  and  ap'  will  be 

M  COS  8 


The  difference  of  phase  of  the  other  pair  of  congruent  rays  bpt 
and  bpi  will  be 

A      47r^  cos  8 

AI=  ~^~ 

If  now  A  -  A1  =  -j-  X,  the  interference  phenomena  which  are  pro- 
duced by  each  pair  of  congruent  rays  are  in  opposite  phase;  that 
is,  if  one  pair  would  produce  a  maximum  of  illumination  at  the 
common  focus,  the  other  would  produce  a  minimum.  The 
resulting  sensation  would  be  a  combination  of  the  two.  Hence, 
in  order  that  interference  bands  may  be  observed  by  an  eye  at  P, 
conditions  must  be  so  arranged  that  the  difference  in  the  As  for 
those  pairs  of  congruent  rays  which  come  to  a  focus  at  the  same 
point  on  the  retina,  be  less  than  half  a  wave.  This  may  be 
accomplished  in  two  different  ways:  First,  we  may  observe 
through  a  small  enough  opening,  the  eye  being  focused  on  the 


THE  MICHELSOX  INTERFEROMETER  53 

point  p.  By  this  simple  device  the  clearness  of  the  interference 
phenomena  can  frequently  be  materially  increased.  Second,  by 
making  the  As  the  same  for  every  pair  of  congruent  rays  which 
are  focused  at  the  same  point  on  the  retina.  This  is  accomplished 
by  making  t  =  h,  and  focusing  the  eye  for  parallel  rays.  The 
second  method  is  generally  to  be  preferred  for  reasons  to  be  given 
presently. 

The  exact  investigation  of  the  form  of  the  fringes  is  a  matter 
of  considerable  complexity.  An  approximate  conception  of  their 
appearance  to  an  eye  placed  at  a  point  P  may  be  obtained  from 
equation  (20).  Thus  let  cdef  (Fig.  17)  represent  the  xy  plane, 
and  P'  the  center  of  a  system  of  rectangular  coordinates  whose  axis 
of  y  is  parallel  to  the  intersection  of  the  mirrors.  Then  P'C  =  #, 

P'D  =  y,  tan  i  =  -=-  and  tan  6  =  -—•  Let  tan  <f>  =  k.  Our  equation 
then  becomes 

D  = 


^         j-        j- 

x  u 

Since  -p-  and  -p-  are  small,  this  equation  reduces  to 

O  Ox 


or 

±m    ^^!-2P2. 


This   is   the   equation   of  a  circle  whose  center  is  at  the  point 

/2P2&      \ 

I —^ — ,  0\-    Hence  the  fringes  are  always  approximately  circles 

whose  centers  lie  on  the  axis  of  x  at  a  distance  from  the  origin 
determined  by  the  inclination  of  the  mirrors,  the  difference  of 
optical  path,  and  the  distance  P  to  the  point  of  observation. 

Two  cases  are  of   special   practical   interest.     First,  if  t  =  t09 


54  MANUAL   OF   ADVANCED    OPTICS 

that   is,    if   the   mirrors   are   parallel,   Ic  =  0,   and   our   equation 
reduces  to 


In  this  case  the  center  of  the  circles  lies  at  the  origin  itself  and 
their  radii  are  given  by 


If  the  difference  of  the  optical  paths  between  the  successive 
rings  counted  from  the  center  be  wX,  we  shall  have 

n\  =  2/  -  D, 
or 

D  =  Zt-  n\. 

Hence  the  radii  of  the  successive  rings  are  given  approximately  by 


(21) 


Second,  if  the  intersection  of  the  mirrors  passes  through  the 
origin,   D  =  0  at  that  point.      In  this  case  the  distance  to  the 


center   of    the   circles,       ~    ,  becomes   infinite,  and   the   fringe 

through  the  origin  is  a  straight  line  parallel  to  the  axis  of  y. 
This  particular  fringe,  since  it  is  the  only  absolutely  straight  one, 
serves  as  a  convenient  mark  from  which  to  begin  measurements. 
It  is  called  the  central  fringe  of  the  system. 

If  the  mirror  om2  (Fig.  16)  is  moved  perpendicular  to  itself 
till  it  passes  to  the  other  side  of  the  mirror  omly  D,  since  its 
value  passes  through  0,  changes  sign,  and  the  center  of  the 


circles  passes  from  H  —  ^—  to  --  ^  —  Hence  on  opposite  sides  of 

the    central    fringe  the  curvature  of    the  fringes  has    opposite 
signs;    that  is,  .on    both  sides  they  appear   convex  toward  the 


THE    MICHELSON    INTERFEROMETER 


55 


central  fringe.     This  fact  is  of  great  assistance  in  locating  the 
central  fringe. 

Experiments 

I.  MEASURE  THE  WAVE  LENGTH  OF  SODIUM.  LIGHT 
APPARATUS. — From  the  discussion  above  it  is  evident  that 
the  essential  parts  of  the  interferometer  are  four  plates  of  glass 
arranged  as  shown  in  Figs.  15  and  19,  and  an  arrangement  for 


moving  one  of  the  mirrors  in  a  direction  normal  to  its  surface. 
This  motion  is  effected  by  mounting  the  mirror  D  upon  a  slide, 
which  can  be  moved  by  a  screw  along  the  ways  EF.  These  ways 
must  be  accurately  straight,  so  that  in  its  motion  the  mirror 
remains  strictly  parallel  to  its  original  position.  The  screw 


56  MANUAL   OF    ADVANCED    OPTICS 

carries  at  its  front  end  a  worm  wheel  M  which  in  turn  is  driven 
by  a  worm  W.  The  worm  can  be  disengaged  from  the  wheel 
when  a  rapid  motion  of  the  screw  is  desired.  The  worm  wheel  is 
graduated  upon  its  front  face. 

The  dividing  plate  A  is  mounted  firmly  in  a  metal  frame 
upon  a  plate  of  brass  H  which  is  screwed  to  one  end  of  the  ways. 
That  side  of  A  which  carries  the  silver  half-film  should  be 
turned  toward  the  plate  B.  This  plate  B  should  be  so  mounted 
upon  the  brass  plate  H  as  to  allow  of  a  small  motion  about  a 
vertical  axis.  This  motion  is  necessary  in  order  to  be  able  to 
adjust  the  two  plates  A  and  B  so  that  they  are  strictly 
parallel.  It  is  also  useful  in  measuring  differences  in  phase  or 
small  fractions  of  a  wave  accurately.  The  mirror  D  is  rigidly 
mounted  upon  the  slide,  while  the  mirror  C  rests  upon  three 
adjusting  screws,  which  are  set  in  a  plate,  which  is  perpendicular  . 
to  the  plate  H  and  firmly  fastened  to  it.  Small  springs  hold  this 
mirror  in  place  against  the  three  screws. 

ADJUSTMENTS. — Measure  roughly  the  distance  from  the  sil- 
ver half -film  upon  the  r.ear  of  the  plate  A  to  the  front  of  the 
mirror  C.  Set  the  mirror  Z>,  by  turning  the  worm  wheel  Jf,  so 
that  its  distance  from  the  rear  of  A  is  the  same  as  that  of  C  from 
A.  This  need  not  be  done  accurately.  It  is  suggested  because  it 
is  easier  to  find  the  fringes  when  the  distance  between  the  mirror 
D  and  the  virtual  image  of  the  mirror  C  is  small.  This  distance 
will  hereafter  be  called  the  distance  between  the  mirrors. 

Now  place  a  sodium  burner,  or  some  other  source  of  monochro- 
matic light,  at  Z,  in  the  principal  focus  of  a  lens  of  short  focus. 
It  is  not  necessary  that  the  incident  beam  be  strictly  parallel. 
Hold  some  small  object,  such  as  a  pin  or  the  point  of  a  pencil, 
between  L  and  A.  On  looking  into  the  instrument  from  0, 
three  images  of  the  small  object  will  be  seen.  One  image  is 
formed  by  reflection  at  the  front  surfaces  of  A  and  Z>;  the  second 
is  formed  by  the  reflection  at  the  rear  surface  of  A  and  the  front 


THE  MICHELSON  INTERFEROMETER  57 

surface  of  D\  the  third  is  formed  by  reflection  from  the  front 
surface  of  C  and  the  rear  surface  of  A.  Interference  fringes  in 
monochromatic  light  are  found  by  bringing  this  third  image  into 
coincidence  with  either  of  the  other  two  by  means  of  the  adjusting 
screws  upon  which  the  mirror  C  rests.  If,  however,  it  is  desired 
to  find  the  fringes  in  white  light,  the  second  and  third  of  these 
images  should  be  brought  into  coincidence,  because  then  the  two 
paths  of  the  light  in  the  instrument  are  symmetrical,  i.e.,  each  is 
made  up  of  a  given  distance  in  air  and  a  given  thickness  of  glass. 
When  the  paths  are  symmetrical,  the  fringes  are  always  approxi- 
mately arcs  of  circles  as  described  above.  If,  however,  the  first 
and  third  images  are  made  to  coincide,  then  the  two  optical  paths 
are  unsymmetrical,  i.e.,  the  path  from  A  to  C  has  more  glass  in 
it  than  that  from  A  to  D,  and  in  this  case  the  fringes  may  be 
ellipses  or  equilateral  hyperbolae,  because  of  the  astigmatism  which 
is  introduced  by  the  two  plates  A  and  B.  It  is  quite  probable 
that  the  fringes  will  not  appear  when  the  two  images  of  the 
small  object  seem  to  have  been  brought  into  coincidence.  This 
is  simply  due  to  the  fact  that  the  eye  can  not  judge  with  sufficient 
accuracy  for  this  purpose  when  the  two  are  really  superposed.  To 
find  the  fringes  then  it  is  only  necessary  to  move  the  adjusting 
screws  slightly  back  and  forth.  As  the  instrument  has  been  here 
described,  the  second  image  lies  to  the  right  of  the  first. 

Having  found  the  fringes  the  student  should  practice  adjust- 
ment until  he  can  produce  at  will  the  various  forms  of  fringes 
described  on  page  54.  Thus  the  circles  appear  when  the  dis- 
tance between  the  mirrors  is  not  zero,  and  when  the  mirror  D 
is  strictly  parallel  to  the  virtual  image  of  C.  The  accuracy  of 
this  adjustment  may  be  tested  by  moving  the  eye  sideways  and 
up  and  down  while  looking  at  the  circles.  If  the  adjustment  is 
correct,  any  given  circle  will  not  change  its  diameter,  as  the  eye  is 
thus  moved.  To  be  sure,  the  circles  appear  to  move  across  the 
plates  because  their  center  is  at  the  foot  of  the  perpendicular 


58  MANUAL    OF    ADVANCED    OPTICS 

dropped  from  the  eye  to  the  mirror  Z>,  but  their  apparent  diam- 
eters are  independent  of  the  lateral  motion  of  the  eye.  For  this 
reason  it  is  advisable  to  use  the  circular  fringes  whenever  possible. 

To  find  the  fringes  in  white  light,  adjust  so  that  the  monochro- 
matic fringes  are  arcs  of  circles.  Move  the  carriage  rapidly  by 
intervals  of  a  quarter  turn  or  so  of  the  wheel  M.  When  the  region 
of  the  white-light  fringes  has  been  passed,  the  curvature  of  the 
fringes  will  have  changed  sign,  i.e.,  if  the  fringes  were  convex 
toward  the  right,  they  will  now  be  convex  toward  the  left.  Having 
thus  located  within  rather  narrow  limits  the  position  of  the  mir- 
ror Z),  which  corresponds  to  zero  difference  of  path,  it  is  only 
necessary  to  replace  the  sodium  light  by  a  source  of  white  light, 
and  move  the  mirror  D  by  means  of  the  worm  slowly  through  this 
region  until  the  fringes  appear. 

These  white-light  fringes  are  strongly  colored  with  the  colors 
of  Newton's  rings.  The  central  fringe, — the  one  which  indi- 
cates exactly  the  position  of  zero  difference  of  path, — is,  as  in 
the  case  of  Newton's  rings,  black.  This  black  fringe  will  be 
entirely  free  from  color,  i.e.,  perfectly  achromatic,  if  the  plates 
A  and  B  are  of  the  same  piece  of  glass,  are  equally  thick,  and 
are  strictly  parallel.  If  they  are  matched  plates,  i.e.,  if  they  are 
made  of  the  same  piece  of  glass  and  have  the  same  thickness, 
their  parallelism  should  be  adjusted  until  the  central  fringe  of 
the  system  is  perfectly  achromatic.  When  this  is  correctly  done, 
the  colors  of  the  bands  on  either  side  of  the  central  one  will  be 
symmetrically  arranged  with  respect  to  the  central  black  fringe. 

MEASUREMENTS. — An  accurate  scale  graduated  to  tenths  of  a 
millimeter  is  set  upon  the  slide  behind  the  mirror  D.  Over  this 
a  micrometer  microscope  is  placed  and  focused  on  the  scale.  The 
microscope  should  be  rigidly  attached  to  the  base  of  the  instru- 
ment so  that  it  does  not  move  relatively  to  the  interferometer  dur- 
ing the  observation.  The  cross-hair  of  the  micrometer  is  set  upon 
one  of  the  tenth  millimeter  divisions.  The  mirror  D  is  then 


THE    MICHELSON    INTERFEROMETER  59 

slowly  moved  with  the  worm  FT,  and  the  number  of  fringes  which 
pass  when  the  cross-hair  of  the  microscope  moves  over  one-tenth 
of  a  millimeter  are  counted.  The  circular  fringes  should  be  used 
because,  as  stated  above,  their  phase  is  independent  of  the  position 
of  the  eye,  so  that  if  the  eye  moves  during  the  observation,  no 
error  will  be  introduced.  •  It  will  be  necessary  to  look  from  time 
to  time  through  the  microscope  so  as  to  note  when  the  cross-hair 
reaches  the  next  tenth  millimeter  mark.  Since  a  motion  of  the 
mirror  of  0.1  mm.  introduces  a  difference  of  path  of  about  340 
waves,  it  is  safe  to  count  300  without  looking  at  the  microscope. 
Having  obtained  the  number  of  fringes  which  pass  when  the  mir- 
ror D  moves  through  0.1  mm.,  then,  since  the  difference  in  path 
introduced  by  this  motion  is  0.2  mm.,  the  wave  length  sought  is 

0  2 
A  =  -^  mm.,  in  which  N  denotes  the  number  of  fringes  counted. 

EXAMPLE 

Sodium  light  was  used  as  a  source  and  the  number  of  waves 
which  passed  while  the  carriage  moved  0.3  mm.  were  counted. 

This  number  was  found  to  be  1018.     Hence 

• 

A  =  T^     Pim.  =  5894  -   10"7  mm. 


II.    DETERMINE   THE  RATIO  OF   THE   WAVE    LENGTHS  OF   THE 
SODIUM  LINES  Dl  AND  D2 

Apparatus  and  adjustment  as  in  Experiment  I. 

MEASUREMENTS.  —  Since  the  yellow  radiation  of  sodium  con- 
tains two  vibrations  of  different  periods,  two  different  sets  of 
fringes  will  be  formed  by  it.  As  the  mirror  D  is  moved  these  sets 
of  fringes,  since  they  are  formed  by  waves  of  different  lengths, 
will  move  at  different  rates.  Thus  in  certain  positions  of  the 
mirror  D  the  bright  fringes  of  one  set  will  fall  upon  the  dark 
fringes  of  the  other  set,  and  the  field  of  view  will  be  almost  evenly 
illuminated.  The  fringes  do  not  disappear  entirely  because  the 


60  MANUAL   OF    ADVANCED    OPTICS 

light  produced  by  the  shorter  of  the  waves  is  more  intense 
than  that  produced  by  the  longer.  At  certain  other  positions 
of  the  mirror  Z>,  the  bright  fringes  of  one  set  will  coincide  with 
the  bright  fringes  of  the  other  set,  and  there  will  be  strong  con- 
trast between  the  bright  and  the  dark  fringes  in  the  field  of  view. 
This  contrast  in  the  intensity  of  the  fringes  is  called  their  visi- 
bility. This  subject  of  visibility  is  treated  at  length  in  the  next 
chapter.  Here  it  is  sufficient  to  note  that  in  the  interval  between 
two  positions  of  greatest  contrast  of  the  fringes,  i.e.,  between  two 
positions  of  maximum  visibility,  there  must  be  one  more  of  the 
shorter  waves  than  of  the  longer.  In  order  to  determine  the  ratio 
of  the  wave  lengths,  then,  it  is  necessary  to  measure  the  distance 
the  mirror  D  moves  in  passing  between  two  positions  of  maximum 
or  of  minimum  visibility.  If  this  distance  be  divided  by  the 
shorter  wave  length  we  obtain  the  number  of  shorter  waves  in  the 
interval.  This  number  minus  one  will  be  the  number  of  longer 
waves  in  the  interval,  and  the  ratio  of  the  wave  lengths  will  be 
the  inverse  of  the  ratio  of  the  number  of  waves. 

In  making  the  observations  it  will  be  found  impossible  to 
determine  accurately  the  position  of  any  one  maximum  or  mini- 
mum. An  accurate  result  may,  however,  be  obtained  in  the  fol- 
lowing way:  Draw  the  mirror  D  as  far  forward  as  is  possible 
without  causing  the  fringes  to  disappear  entirely.  Then  move  the' 
mirror  backward  by  jumps  of  about  one-twentieth  of  a  turn  of  the 
worm  wheel.  Take  the  reading  on  the  worm  wheel  at  the  points 
which  appear  to  be  either  maxima  or  minima.  In  this  way  about 
twenty  readings  of  the  positions  of  the  maxima  and  twenty  of 
those  of  the  minima  can  be  obtained.  The  average  should  then 
be  taken  by  subtracting  the  first  reading  in  each  set  from  the 
eleventh,  the  second  from  the  twelfth,  etc.,  and  then  taking  the 
mean  of  these  averages.  In  making  the  calculation  it  is  to  be 
noted  that  the  difference  of  path  is  twice  the  distance  through 
which  the  mirror  has  moved. 


THE    MICHELSON   INTERFEROMETER  61 

EXAMPLE 

The  distance  between  the  positions  of  the  maximum  clearness 
of  the  fringes  was  determined  in  the  way  described  above.  In  all, 
thirty  maxima  and  thirty  minima  were  read.  The  mean  value  of 
the  interval  was  found  to  be  0.5802  mm.  Hence 

0.5802 


005890 

iii  =  n-2  -  1  =  984, 

A!      085 
A,  =984' 

in  which  A^  is  the  wave  length  of^D^  and  \%  that  of  Dz. 

If  we  wish  the  difference  between  the  wave  lengths  in  milli- 
meters we  may  proceed  thus  : 

.5802      .5802  11  1 

=  1,    or     --  -  = 


A,  X,  A,     A!      .5802 

Therefore,     A:- A,,  =  -——-=  5.98  •  10~7  mm. 


III.  DETERMINE    THE    INDEX   OF    REFRACTION    AND  THE  DIS- 
PERSION OF  A  PIECE  OF  GLASS 

Apparatus  and  adjustment  as  in  Experiment  I. 

MEASUREMENTS. — In  order  to  perform  this  experiment  in  a 
theoretically  rigorous  way  there  should  be  added  to  the  interferom- 
eter two  extra  metal  frames  similar  to  those  used  to  hold  the 
mirrors,  one  in  front  of  each  of  the  mirrors  D  and  C.  These 
frames  should  be  set  upon  pivots  so  that  they  can  be  rotated 
about  a  vertical  axis.  The  one  in  front  of  the  mirror  C  should  be 
arranged  with  a  worm  wheel  or  a  tangent  screw  so  that  it  can  be 
rotated  slowly  and  steadily.  Two  pieces  of  the  glass  whose  index 
of  refraction  and  dispersion  are  to  be  determined  should  be  used. 
These  pieces  should,  of  course,  have  the  same  thickness.  One  of 


62  MANUAL   OF    ADVANCED    OPTICS 

them  is  waxed  to  each  of  the  movable  frames  so  as  to  cover  half 
—the  same  half — of  the  field  of  view.  The  interferometer 
must  then  be  adjusted  for  the  white-light  fringes.  The  piece 
of  glass  which  is  in  front  of  the  mirror  D  is  then  turned 
through  any  angle,  say  15°  to  20°.  In  this  way  extra  glass  is 
introduced  into  the  path  of  the  light  between  A  and  D.  This 
extra  glass  should  then  be  compensated  for  by  turning  the  other 
piece,  that  between  A  and  (7,  through  the  same  angle.  When  the 
angles  through  which  the  two  pieces  have  been  turned  are  the 
same,  the  white-light  fringes  will  appear  in  the  half  of  the  field  of 
view  which  is  covered  by  the  two  plates.  This  turning  of  the 
second  plate  of  glass  should  be  done  slowly  with  the  worm  wheel, 
and  the  fringes  which  pass  during  the  operation  should  be 
counted.  The  angle  through  which  this  plate  turns  must  be 
measured  by  fastening  to  the  frame  which  carries  it  a  small  mir- 
ror, and  reading  the  angle  through  which  this  mirror  turns  with 
an  ordinary  telescope  and  scale.  Before  measuring  this  angle,  care 
must  be  taken  to  have  the  plate  perpendicular  to  the  beam 
passing  through  it.  This  can  be  done  by  rotating  the  plate 
through  the  position  in  which  it  is  at  right  angles  to  the  beam, 
and  noting  the  point  at  which  the  fringes  reverse  their  direction 
of  motion ;  for  it  is  evident  that  when  the  plate  is  normal  to  the 
beam  its  optical  thickness  is  a  minimum,  and  therefore  a  turning  of 
the  plate  in  either  direction  will  increase  the  optical  thickness,  and 
cause  the  fringes  to  move  in  one  particular  direction.  It  will, 
however,  be  found  that  the  plate  can  be  turned  through  a  consid- 
erable angle  before  the  fringes  move  appreciably.  Therefore,  to 
obtain  the  scale  reading  which  corresponds  to  the  normal  position 
of  the  plate,  turn  the  plate  in  one  direction  until  two  or  three 
fringes  have  passed,  and  take  the  reading  on  the  scale.  Then  turn  it 
in  the  other  direction  until  the  same  number  of  fringes  has  passed, 
and  take  the  reading.4  The  mean  of  these  two  readings  will  then 
be  the  reading  which  corresponds  accurately  to  the  normal  posi- 


THE    MICHELSON    INTERFEROMETER 


63 


tion  of  the  plate.  Having  then  counted  the  fringes  which  pass 
while  the  plate  is  turning  through  the  angle  i,  and  having  meas- 
ured that  angle,  the  index  of  refraction  is  obtained  as  follows:  Let 
t  represent  the  thickness  of  the  plates  of  glass,  and  2JV  the  num- 
ber of  fringes  counted  while  turning  through  the  angle  i.  Let, 
further,  AB  (Fig.  20)  represent  the  direction  of  the  light,  MNOP 


0 


the  plate  in  its  position  perpendicular  to  the  beam,  and  M'N'O'P' 
its  position  after  it  has  been  turned  through  the  angle  i.  Let  the 
two  surfaces  OP  and  O'P'  intersect  at  c,  and  draw  through/,  the 
intersection  of  the  surface  M'N'  with  AB,  the  line//?  parallel  to 
MN.  The  light  incident  along  AB  upon  O'P'  will,  when  the 
plate  has  been  turned,  travel  along  the  path  cgh.  It  is  evident  that 
before  the  turning  the  optical  distance  between  the  planes  OP  and  /ft 

consisted  of  a  distance  ce  =  t  in  glass,  and  a  distance  ef= 1 

J      cos  i 

in  air.     After  the  turning,  the  optical  distance  between  these  two 

i 


planes  consists  of  a  distance  eg 


cos  r 


in  glass,  and  of  a  distance 


7      t  sin  (i  —  r)  ,       .  .      .     .       ,  .  ,       , 

nil  =  — -  tan  i  in  air,  in  which  r  denotes  the  angle  of  refrac- 

cos  r 

tion.    The  numbers  of  waves  in  these  various  distances  are  obtained 
by  dividing  the  distances  in  air  by  A,  the  wave  length,  and  those 

in  glass  by  — ,  in  which  /w.  stands  as  usual  for  the  index  of  refrac- 


64  MANUAL   OF   ADVANCED    OPTICS 

tion.  The  difference  between  the  number  of  waves  in  the  optical 
path  between  the  planes  OP  and  fli  before  the  turning  and  the 
number  in  the  path  after  the  turning  is  half  of  the  number  which 
has  been  counted,  because  in  the  observation  the  light  has  passed 
twice  through  the  plate.  Therefore  the  following  equation  is 
obtained : 

tu.        t  sin  (i  —  r}  .  t 

—!—  -] * '-  tan  %-  tu. ;  + 1  =  JV  A. 

cos  r  cos  r  cos  i 

If  this  equation  be  reduced  with  the  help  of  the  equation  -    —  =  /n, 

JV2A2 
and  solved  for  /*,  there  results,  neglecting  the  term  -—7  -  in  the 

lit 

numerator, 

(;-^X)(l-cos/) 
^     #(l-cos  i)  -  NX 

The  thickness  t  of  the  glass  may  be  measured  with  the  calipers  or 
in  any  other  accurate^way. 

It  was  shown  in  the  chapter  on  the  Fresnel  mirrors  that  if 
the  optical  symmetry  of  the  two  paths  over  which  the  light  travels 
is  disturbed  by  the  introduction  of  a  plate  of  some  transparent 
substance,  the  fringes  in  white  light  no  longer  possess  a  truly 
achromatic  central  fringe,  but  one  which  may  seem  fairly  achro- 
matic, and  which  is  displaced  from  the  true  position  of  the  central 
fringe  of  the  set  of  fringes  which  correspond  to  the  wave  length  A. 
by  a  number  of  fringes  [cf.  equation  (17)] 

P -?= 

in  which  t'  =  — 1.  (23) 

cos  r 

Now,  when  the  glass  which  was  added  to  the  path  AD  by  turn- 
ing the  plate  in  front  of  the  mirror  Z>,  is  compensated  for,  by 
rotating  the  plate  in  front  of  the  mirror  (7,  we  add  extra  glass  to  the 
path  A  C  also.  The  two  paths  are  thus  made  finally  symmetrical, 


THE  MICHELSON  INTERFEROMETER  65 

so  that  the  achromatic  light  fringe  to  which  we  count  indicates  the 
true  position  of  the  central  band  of  the  monochromatic  system.  By 
such  a  count,  then,  we  obtain  the  true  number  of  fringes  through 
which  the  monochromatic  system  has  been  shifted,  namely^?.  It 
is  possible,  however,  to  compensate  for  the  extra  glass  in  the  path 
AD  in  another  way,  namely,  by  drawing  the  mirror  D  toward  A. 
If  we  do  this  and  count  the  fringes  which  pass,  we  count  to  the 
shifted  position  of  the  white-light  fringes,  i.e.,  we  count  the 
number  pr.  If  then  we  make  the  count  both  ways,  once  by  turn- 
ing the  plate  in  front  of  the  mirror  C,  and  once  by  drawing  up 
the  mirror  Z>,  the  plate  in  front  of  C  being  perpendicular  to  the 
beam,  we  determine  both  p  and  p'  of  the  above  equation.  Since 
/  and  X  are  also  known,  it  is  in  this  way  possible  to  determine  the 
B  of  the  Cauchy  dispersion  equation  (p.  39).  With  this  value  of 
B  and  the  value  of  p  determined  from  equation  (22) ,  it  is  then 
possible  to  determine  the  A  of  the  dispersion  equation.  Thus 
with  a  single  source  of  monochromatic  light  it  is  possible  to 
determine  both  the  index  of  refraction  and  the  dispersion  of  a 
plate  of  glass. 

EXAMPLE 

Two  pieces  of  optical  glass  were  mounted  in  the  instrument 
as  described  above.  The  thickness  of  the  glass  was  6.81  mm., 
i.e.,  t  =  6.81  mm.  One  of  the  plates  was  then  turned  through  an 
angle  «,  which  was  measured  with  a  telescope  and  scale,  and  found 
to  be  i  =  16°  41'  30".  The  other  piece  was  then  turned,  and  the 
sodium  fringes  counted  until  the  fringes  in  white  light  appeared 
over  the  whole  field.  This  number  was  342.  Since  the  light 
traverses  the  plates  twice,  the  number  N  in  the  formula  is  half  of 
this,  i.e.,  N=  171.  Hence  the  index  of  refraction  for  sodium  is 
jt.va  =  1.5180.  The  compensating  plate  was  then  turned  back  till 
it  was  normal  to  the  path  of  the  light,  and,  the  first  piece  of  glass 
remaining  inclined  at  the  angle  i,  the  movable  mirror  was  drawn 


66  MANUAL    OF   ADVANCED    OPTICS 

up  and  the  fringes  again  counted  until  the  white  fringes  appeared 
in  their  first  position.     This  number  of  fringes  was  %p'  =  355. 
Hence  %(p'-p)  =  355-342  =  13  or/ -^  =  6. 5. 

From  equation  (23),  t'  =  .1254  mm.     Hence  from  equation  (17) 

£  =  53  •  10-10, 
and  therefore  (cf.  p.  39),  A  =  1.5027. 

With  these  values  of  A  and  B  the  index  of  refraction  for  the 
green  line  of  mercury  was  calculated  (A  =  5461  •  10~7).  The  result 
was  fAHg  =  1.5205.  The  fringes  were  then  counted  in  mercury 
light,  and  the  result  was  2^=370.  With  this  N  we  get  from 
equation  (22)  pHg  =  1.5204. 

IV.  DETERMINE  THE  CHANGE  or   PHASE   PRODUCED   BY   PER- 
PENDICULAR REFLECTION  AT  A  SILVER  SURFACE 

Apparatus  and  adjustments  as  in  Experiment  I. 

MEASUREMENTS. — One  of  the  mirrors  C  or  D  of  the  interferom- 
eter must  be  removed  and  freshly  silvered  over  three-quarters  of 
its  surface.  This  is  best  accomplished  by  keeping  one-half  of  the 
mirror  covered  with  a  piece  of  glass  while  it  is  in  the  silvering 
solution.  When  the  silver  is  sufficiently  deposited  so  that  the 
film  is  perfectly  opaque,  the  solution  is  poured  off,  the  mirror  and 
the  tray  which  holds  it  rinsed  with  distilled  water,  the  piece  of 
glass  on  the  surface  of  the  mirror  turned  through  90°,  and  a  fresh 
solution  poured  on.  In  this  way  the  surface  of  the  mirror  is 
coated  with  silver  over  three-quarters  of  its  surface,  as  shown  in 
Fig.  21.  The  quarter  marked  a  has  upon  it  two  layers  of  silver, 
~b  and  c  each  has  one  layer,  and  d  has  none.  If  now  the  mirror 
be  replaced  in  the  interferometer,  and  the  fringes  found,  it  will  be 
noted  that  where  the  fringes  cross  the  boundaries  of  these  four 
sections  of  the  surface,  they  are  displaced.  Thus  a  fringe  which 
passes  from  a  to  I  will  not  be  a  straight  line  or  an  arc  of  a  circle, 
but  that  portion  of  it  which  is  over  the  surface  a  will  be  displaced 


THE    MICHELSON    INTERFEROMETER 


67 


with  respect  to  that  over  the  surface  b  by  an  amount  which 
depends  upon  the  thickness  of  the  film  ac.  If  this  displacement 
is  measured,  we  thereby  determine  the  thickness  of  the  film  ac. 


FIGURE  21 

In  order  to  measure  it,  the  white-light  fringes  must  first  be  found 
so  as  to  determine,  by  means  of  the  central  black  fringe,  in  which 
direction  the  shifting  has  taken  place.  Having  thus  determined 
the  direction  of  the  shifting,  its  amount  is  measured  by  the  com- 
pensator. There  'should  be  fastened  to  one  end  of  the  frame 
which  holds  the  compensator,  a  small  spiral  spring.  The  other 
end  of  the  spring  should  be  fastened  to  a  string  which  may  be 
wound  about  a  pin.  The  pin  must  carry  a  graduated  drum  or 
circle,  so  that  its  position  may  be  read  and  thereby  the  tension  of 
the  spiral  spring  determined.  The  tension  of  this  spring  is 
opposed  by  the  elasticity  of  the  stud  by  which  the  compensator 
frame  is  fastened  to  the  plate  H.  If  the  pin  is  turned  so  as  to 
tighten  or  loosen  the  spiral  spring,  the  compensator  will  turn 
through  a  small  angle,  and  this  angle  will  be  proportional  to  the 
amount  of  the  turning  of  the  pin.  To  measure  the  difference  of 
phase,  the  pin  is  turned  until  one  fringe  has  passed,  and  the 
angle  through  which  the  pin  has  turned  is  read  upon  the 
graduated  head.  The  pin  should  then  be  turned  until  the 
shifted  part  of  the  fringe  comes  to  the  position  of  the  unshifted 
part,  and  the  angle  through  which  it  has  turned  is  read  upon  the 
graduated  head.  The  ratio  of  the  angles  through  which  the  pin 
has  turned  in  these  two  operations  is  then  the  fraction  of  a  wave 


68  MANUAL   OF   ADVANCED    OPTICS 

by  which  the  fringe  is  shifted,  and  half  of  this  fraction  is  the 
thickness  of  the  film  ac  in  wave  lengths.  Of  course  this  measure- 
ment should  be  made  in  monochromatic  light,  the  white-light 
fringes  being  used  merely  to  recognize  the  direction  of  the  shift. 
Having  thus  measured  the  thickness  of  the  film  ac,  it  is  only 
necessary  to  measure  the  shift  in  a  fringe  at  the  junction  between 
c  and  d,  in  order  to  be  able  to  calculate  the  change  of  phase  due 
to  the  reflection  at  the  silver  surface.  In  calculating  this  change 
of  phase,  account  must  be  taken  of  the  fact  that  half  a  wave  is 
lost  at  the  reflection  upon  glass,  and  also  of  the  direction  of  the 
shift  after  allowance  has  been  made  for  the  thickness  of  the  film 
ac.  If  the  shift  is  in  the  same  direction  as  that  in  the  fringe 
across  ab,  then,  since  a  is  nearer  the  observer  than  J,  the  wave 
from  the  silver  surface  is  ahead  of  that  from  the  glass. 

The  difference  of  phase  between  the  light  reflected  at  a  surface 
glass-air,  and  that  reflected  at  a  surface  glass-silver,  is  very  easily 
obtained  with  the  aid  of  a  plane  parallel  plate  of  optical  glass 
which  has  been  silvered  over  half  of  one  surface.  This  plate 
must  be  introduced  into  the  interferometer  in  place  of  the  mirror 
D,  with  the  silver  side  away  from  the  observer.  A  second  piece 
of  the  same  plate  of  glass  must  then  be  introduced  in  front  of  the 
mirror  C  in  order  to  compensate  for  the  extra  glass  added  by  the 
introduction  of  the  first  plate  into  the  path  AD.  The  white-light 
fringes  having  been  found  upon  the  rear  of  the  plate  in  front  of 
D,  and  been  set  perpendicular  to  the  dividing  line  between  the 
silvered  and  the  unsilvered  portions  of  the  plate,  the  displacement 
of  the  central  fringe  is  measured  as  described  above.  Since  the 
light  is  reflected  from  the  glass-air  surface  without  change  of 
phase,  the  shifting  of  the  fringe  indicates  a  retardation,  i.e.,  a 
loss  of  part  of  a  wave. 

EXAMPLES 

1.  One  of  the  mirrors  of  the  interferometer  was  coated  with  a 
double  film  of  silver  as  illustrated  in  Fig.  21.  The  displacement 


THE   MICHELSON    INTERFEROMETER  69 

of  the  fringe  across  ab  was  measured  in  sodium  light  and  found 
to  be  0.26.  Hence  the  thickness  of  the  film  ac  was  0.13X.  The 
fringe  across  cd  was  displaced  0.17  of  a  fringe.  Since  the  reflec- 
tion upon  the  glass  surface  d  produces  a  change  of  phase  of  0.5  of 
a  wave,  the  retardation  produced  hy  the  silver  is  0.5  +  0.17  —  0.26 
=  0.41X. 

2.  The  displacement  of  the  fringe  on  the  glass-silver  surface 
was  found  to  be  0.28A. 

For  further  study,  of  the  applications  which  can  be  made  of  the  inter- 
ferometer the  student  is  referred  to  the  following: 

Michelson  and  Morley,  "On  the  Relative  Motion  of  the  Earth  and 
Ether,"  Am.  Jour.  Sci.  (3)  22,  p.  120,  1881;  34,  p.  333,  1887;  Phil.  Mag.  (5) 
£4,  p.  449,  1887.  "On  the  Effect  of  the  Motion  of  the  Medium  upon  the 
Velocity  of  Light,"  Am.  Jour.  Sci.  (3)  31,  p.  377,  1886.  "On  a  Method  of 
Using  the  "Wave  Length  of  Sodium  Light  as  a  Practical  Standard  of 
Length,"  Am.  Jour.  Sci.  (3)  &#,  427,  1887;  Phil.  Mag.  (5)  24,  p.  463;  Am. 
Jour.  Sci.  (3)  37,  p.  181,  1889. 

Michelson,  "Light  Waves  and  Their  Applications  to  Meteorology," 
Nature,  49,  p.  56,  1893.  "Valeur  du  Metre  en  longeurs  d'ondes  lumin- 
euse,"  Trav.  et  Mem.  Bur.  Internat.  Poids  et  Mes.  XI,  p.  1,  1894;  "On  the 
Relative  Motion  of  the  Earth  and  Ether,"  Am.  Jour.  Sci.  (4)  3,  p.  475, 1897. 

Morley  and  Rogers,  "On  the  Measurement  of  the  Expansion  of  Metals 
by  the  Interferential  Method,"  Phys.  Rev.  4,  pp.  1  and  106,  1896. 

Wads  worth,  "On  the  Application  of  the  Interferometer  to  the  Meas- 
urement of  Small  Deflections  of  a  Suspended  System,"  Phys.  Rev.  4, 
p.  480,  1897. 

Hull,  "On  the  Use  of  the  Interferometer  in  the  Study  of  Electric 
Waves,"  Phys.  Rev.  5,  p.  231,  1897. 

Johonnott,  "On  the  Thickness  of  the  Black  Spot  on  Liquid  Films," 
Phil.  Mag.  (5)  47,  p.  501,  1899. 

Earhart,  "On  Sparking  Distances  between  Plates,"  Phil.  Mag.  (6)  1, 
p.  147,  1901. 

Gale,  "On  the  Relation  between  Density  and  Index  of  Refraction  of 
Air,"  Phys.  Rev.  14,  p.  1,  1902. 


VI 

THE   VISIBILITY    CURVES 
Theory 

In  Chapter  II  it  has  been  shown  that  it  is  possible  to  deter- 
mine the  width  of  a  rectangular  source  and  the  distance  between 
two  such  sources  by  observations  made  with  the  double  slit.  In 
these  experiments  the  fringes  disappeared  when  the  distance 
between  the  slits  was  such  that  the  angle  subtended  by  the 
sources  was  one  wave  divided  by  that  distance,  and  also  when 
that  distance  was  such  that  the  angular  width  of  each  single 
source  was  equal  to  a  wave  length  divided  by  it.  In  Chapter  III, 
the  two  slits  have  been  converted  into  an  interferometer,  and  in 
Chapter  V  we  have  used  the  interferometer  to  measure  the  ratio 
of  the  wave  lengths  of  the  two  sodium  lines,  by  determining  the 
number  of  waves  in  the  change  which  takes  place  in  the  distance 
between  the  mirrors  in  passing  from  one  position  of  maximum 
visibility  to  the  next.  The  close  similarity  between  the  two 
experiments  must  be  evident  at  once, — the  difference  lying  in  the 
fact  that  in  the  case  of  the  two  slits  we  have  angles  to  resolve, 
while  with  the  interferometer  we  have  differences  in  wave  lengths, 
or  rather  in  numbers  of  vibrations,  to  determine.  The  resemblance 
becomes  even  closer  if  we  conceive  the  spectral  source  to  be 
resolved  as  far  as  possible  by  an  ordinary  spectroscope.  The 
sodium  lines,  for  example,  would  then  appear  as  two  line  sources, 
i.e.,  they  would  very  much  resemble  the  double  source  consisting 
of  a  pair  of  parallel  slits  as  treated  above. 

We  might  expect  then  that  the  equations  which  connect  the 
visibility  curves  with  the  distribution  of  light  in  the  source  would 

70 


THE    VISIBILITY    CURVES  71 

be  very  similar  in  the  two  cases,  "and  would  differ  only  in  the  fact 
that  angles  in  the  case  of  the  two  slits  would  be  replaced  in  the 
case  of  the  interferometer  by  wave  lengths  or  numbers  of  vibra- 
tions. 

The  solution  of  a  visibility  curve  is  very  difficult.  It  will  help 
us  much  in  obtaining  such  solutions  if  we  begin  by  the  inverse 
process  of  assuming  a  known  distribution  of  light  and  plotting  the 
corresponding  visibility  curve.  Fig.  22  gives  a  series  of  such 
curves.  The  nature  of  the  distribution  in  the  source  is  shown  at 
the  left,  and  the  actual  vibrations  are  plotted,  the  visibility  curve 
being  the  envelope  of  the  curve.  The  abscissae  of  the  curves  rep- 
resent distance  traveled,  and  the  ordinates  intensity.  Thus  in 
Fig.  22  the  curve  1  represents  the  resultant  of  two  trains  of 
homogeneous  waves  of  the  same  amplitude  but  with  slightly 
different  periods  which  start  in  the  same  phase  at  a.  When 
they  have  traveled  a  distance  ab,  they  are  seen  to  be  in  opposite 
phase,  and  the  visibility  curve  comes  to  zero.  It  is  quite  clear 
that  the  distance  they  have  to  travel  before  they  come  into 
opposite  phases  depends  upon  the  difference  of  their  periods. 
So  we  can  already  guess  that  a  determination  of  the  distance  ab 
would  lead  to  some  knowledge  of  that  difference  in  the  periods. 

In  the  case  of  the  interferometer  we  have  formed  by  the  two 
trains  of  waves  two  separate  sets  of  fringes,  and  when  the  movable 
mirror  is  displaced,  these  sets  travel  across  the  field  at  different 
rates,  as  was  shown  on  page  59.  When  a  certain  difference  of 
path  has  been  introduced,  represented  by  ab  in  curve  1,  these  two 
sets  of  fringes  overlap  so  as  to  present  an  evenly  illuminated  field 
of  view  and  the  visibility  curve  comes  to  zero.  As  the  difference 
of  path  is  further  increased,  the  fringes  soon  come  into  such  posi- 
tions that  one  set  has  overtaken  the  other  by  one  whole  fringe, 
and  then  we  have  a  maximum  of  visibility  as  indicated  at  c.  Thus 
if  we  interpret  the  vibrations  which  unite  to  form  the  curve  1  as 
fringes,  i.e.,  as  periodic  variations  of  intensity,  and  consider  that 


72 


MANUAL    OF    ADVANCED    OPTICS 


the  distance  traveled  is  replaced  in  the  interferometer  by  differ- 
ence of  path,  then  the  envelope  represents  the  variations  in  the 
visibility  of  the  resultant  set  of  fringes  as  the  two  separate  sets 
pass  by  in  the  field  of  view.  Hence  the  envelopes  of  the  curves 
in  Fig.  22,  are,  in  the  case  of  the  interferometer,  the  visibility 


'•A 

L     |™^ — A/WWV — ^ 


5. 


j.    I II    ||to*^  vwwwv^^ 


7. 


1 


FIGURE  22 


curves,  and  from  them  we  can  draw  conclusions  as  to  the  nature  of 
the  source. 

Thus  curve  2  represents  the  visibility  curve  which  corre- 
sponds to  a  double  source,  each  of  whose  components  is  broad, 
i.e.,  does  not  send  out  waves  of  one  definite  period  only,  but  waves 
whose  lengths  vary  between  the  limits  X  and  \  +  rfA.  It  will  be 
noted  that  the  distance  between  the  centers  of  these  two  sources 


THE   VISIBILITY   CURVES  73 

is  the  same  as  that  of  curve  1,  so  that  the  positions  of  zero  visibility 
are  not  changed.  The  effect  of  broadening  the  source  is  seen  to 
be  a  decrease  in  the  visibility  at  each  successive  maximum,  so 
that  the  fringes  soon  disappear  altogether. 

Curve  3  corresponds  to  two  homogeneous  radiations  of 
unequal  amplitudes,  and  curve  4  represents  a  single,  broad,  uni- 
formly illuminated  source.  The  other  curves  are  easily  under- 
stood. 

Let  us  now  take  up  the  analytical  discussion  of  the  subject. 
According  to  equation  (6),  page  22,  the  intensity  of  illumination 
produced  at  any  point  by  two  congruent  rays  of  equal  brightness  is 
expressed  by 

A2  =  ±A*  cos2  TT  -  =  2A?  (l  +  cos  Zir  -  V 

Now  2  A  !2  represents  twice  the  intensity  of  each  of  the  two  rays. 
This  intensity  may  be  regarded  as  a  function  of  the  wave  length, 
so  that  we  may  replace  %A  *  by  $  (X)  and  our  equation  becomes 

/A  =  »A(X)  +  <A  (X)    COS27T^. 

Since  each  of  the  separate  vibrations  is  independent,  the 
resultant  intensity  /  will  be  the  integral  of  this  expression  taken 
between  the  limits  Xx  and  X-j.  Let  now  X<,  represent  a  wave  length 
which  is  intermediate  between  Xj  and  X2.  Then 


in  which  p  represents  the  number  of  waves  in  the  distance  8. 
The  intensity  may  then  be  regarded  as  a  function  of  x  between 
the  limits  —  xl  and  +  x2.  Hence  we  may  replace  ^  (X)  d\  by 
<£  (x)  dx,  and  there  results  for  the  total  intensity 

x)  dx  +       (x)  cos  2/?7T  (1  +#)  dx. 


74  MANUAL    OF    ADVANCED    OPTICS 

If  now  we  expand  this  cosine  and  introduce  the  notation, 


C=J<}>  (x)  cos  %pirxdx 

S  =  /<£  (a;)  sin  %pirxdx, 

there  results 

/  =  P  +  C  cos  2/jrr  —  S  sin  2/?ir. 

Now  as  long  as  the  interval  from  Xi  to  x2  is  small,  the  variation 
of  the  values  of  C  and  S  corresponding  to  a  change  in  p  is  small. 
Hence  as  on  page  24,  the  maxima  and  minima  of  /  are  deter- 
mined by  y-  =  0,  i.e.,  by 

C  sin  2jt?7r  +  S  cos  2jt??r  =  0. 
Hence 

I=P  ±  \/C'z  +  S2. 

The  visibility  may  then  be  denned  by  equations  (7),  (8),  and 
(9),  page  25.  Thus  in  the  case  of  a  single  uniformly  illuminated 
source  <£  (x)  =  const.,  which  sends  out  waves  for  which  x  varies 

within  the  limits  ±  —  >  we  have 


a 

\°r 

L   1-4    1 


sn  nira 


V=-     -±~-.|ocftft*mfe. 

£    a  1  pwa 


HI 


as  on  page  25,  but  in  this  case  p  is  equal  to  the  number  of  waves 
in  the  difference  of  path,  and  a  is  a  small  fraction  which  deter- 
mines the  width  of  the  source.  It  will  be  noted  that  the  visibility 
is  equal  to  zero  when  pa  =  1,  2,  3,  etc.,  i.e.,  when 

123 

p  =  -,  —  ,  —  ,   etc. 
*     a'  a'  a' 


THE   VISIBILITY    CURVES  75 

The  more  important  case  is  that  in  which  the  distribution  of 
light  in  the  source  is  represented  by 

<»  (x)  =  *-*"', 

which  distribution  is  in  accord  with  Maxwell's  Law  deduced  from 
the  theory  of  probability. 

In  this  case  when  k  is  large  the  value  of  the  integral  diminishes 
rapidly  with  increasing  x,  the  terms  near  the  origin  being  the  only 
important  ones.  Hence  the  limits  of  the  integration  may  be 

taken  as  ±  x.     Then 

I 

['- '              (      e  '"***  cos 
J-~   _  «/-« _  p     yfc2 

*  r    —  : —  —  ~^~r. —  V 


It  will  be  noted  that  the  curve  is  not  periodic,  but  diminishes 
gradually  as  p  increases.  If  we  assume  that  the  source  is  prac- 
tically limited  when  <£  (.?;)  has  reached  a  value  equal  to  one-half  of  its 

maximum  value,  then,  calling  —  the  corresponding  value  of  #, 
we  have 

I  

-  =  e       4  ,       IV 
Hence 


If  now  q  represent  the  value  of  p  for  which  V  •=•  —  ,  then 


hence 

a_  _  log  2  _  .22 

2    =    "~    =          ' 


76  MANUAL   OF    ADVANCED    OPTICS 

If  this  value  of  a  be  substituted  above,  there  results 

ff8  log  2 

V=e        v*    , 
or 


It  will  be  noted  that  —  is  a  small  fraction  denoting  parts  of  a  wave 
& 

length.  If  we  wish  to  get  the  width  of  the  source  in  millimeters 
we  must  multiply  by  X.  Also  both  p  and  q  are  numbers  of  waves. 
If  we  wish  to  have  them  expressed  in  millimeters  we  multiply 
both  by  X  expressed  in  millimeters.  Thus  letting  p\  =  JT, 
q\  =  A,  we  get 

F-a-f,       y-fV  \   ,  (34) 

in  which  W  represents  the  width  of  the  source  in  millimeters. 

It  has  been  shown  on  page  26,  that  the  equation  of  the 
visibility  curve  for  a  double  source  differs  from  that  for  a  single 
source  by  the  addition  of  a  cosine  factor.  In  general  let  us 
suppose  we  have  a  series  of  similar  sources  which  lie  about  the 
origin  of  coordinates  and  whose  distribution  of  intensity  is 
expressed  by  <f>  (#).  The  expression  for  the  number  of  vibrations 
of  the  waves  of  any  source  may  be  put  in  the  form 

1       1 

l-X<*#*>v 

For  a  symmetrical  distribution   (5=0),  the  integrals  P  and   C 
take  the  form 

f<f>  (x)  sin  %pir  (d  +  x)  =  C  sin  Zpird, 

J  <j>  (x)  cos  Zpn  (d  -f  x)  =  C  cos  2pird. 

Hence  if  U  represent  the  visibility  which  results  from  all  the 
sources 

2  _  (Sff  sin  %mf2  +   2C  cos  2W2 


THE    VISIBILITY    CURVES  77 

Now  the  visibility  due  to  each  source  by  itself  is  represented  by 

r=  — 

P ' 
therefore 


_  (2  VP  sin  Zjnrd)*  +  (S  VP  cos  Zpwd)* 


or 

W  cos  2pir  (d'  —  d) 


If  the  law  of  the  distribution  of  the  light  in  the  separate 
sources  is  the  same,  while  their  intensities  are  proportional  to 
factors  r,  /,  r",  etc.,  the  visibility  V  produced  by  each  will  be 
the  same,  but  P  will  be  proportional  to  r ;  hence,  for  this  case 


_ 


cos  %pir  (df  -  d)     2 


In  the  case  of  a  double  source,    the  ratio  of  the  intensities  of 
whose  components  is  r  :  1,  this  reduces,  if  d0  =  d'  —  d,  to 


!  1  +  r*  +  2r  cos  0  (     . 

(1  +  r)2 

and  if  the  two  sources  have  equal  intensities,  to 

U=  Vcospirt10.  (26) 

The  value  of  d0  may  be  found  from  the  positions  of  zero  visibility, 
i.e.,  the  points  at  which  U  '=  0.  In  case  the  two  lines  have  not 
equal  intensities,  we  may  still  determine  the  value  of  d0  from  the 
period  of  the  curve.  Thus  in  the  case  of  the  two  sodium  lines 
the  visibility  curve  reaches  its  first  minimum  when/;  =  492  waves; 

13  1 

hence,  since  U  =  0  when  pd0  =  -•>  —•>  etc.  ,  rf0  =  T^'    So  it  follows 


that  do  represents  a  fraction  of  a  wave  length,  so  that  when  it  is 
multiplied  by  AQ  we  obtain  the  difference  between  the  wave  lengths 
of  the  two  sources  in  millimeters,  i.e.,  A^  -\z  = 


78  MANUAL    OF    ADVANCED    OPTICS 

Since  the  period  of  the  curve  is  the  distance  between  two  suc- 
cessive minima  which  correspond  to  differences  of  path  pl  and  p^ 
we  have,  denoting  the  number  of  waves  in  that  period  by  p^ 


but   if   D   represents  the  length  of    the  period   in  millimeters, 

=  A,-V  (27) 


p0  =  —-  ,  and  therefore 


In  general  p  -  ~^—->  hence  equation  (25)  becomes 

A 

Y 

cos  2ir 


Since  this  coefficient  of  V  appears  frequently  it  will  be  denoted 
symbolically  by  cos  -=j- 

The  inverse  problem  of  determining  the  form  of  the  distribu- 
tion when  the  visibility  curve  is  given  is  more  difficult.  The 
general  solution  is  shown  by  Kayleigh*  to  depend  upon  both  C 
and  S.  Now  the  visibility  curve  alone  determines  only  C2  +  S*. 
Hence  the  solution  is  not  single  valued  unless  we  can  obtain  a 
second  relationship  between  C  and  S.  This  can  be  done  by 
determining  the  displacement  of  the  phase  of  the  fringes  as  the 
difference  in  path  is  increased.  We  may,  however,  obtain  a  fairly 
accurate  idea  of  the  distribution  in  quite  a  number  of  cases  from 
the  visibility  curve  alone  if  we  assume  that  we  are  dealing  with  a 
source  in  which  the  distribution  is  symmetrical,  for  in  this  case 
S  =  0,  and  the  solution  is  definite. 

The  process  of  determining  the  visibility  curve  of  a  given 
source  by  observation  is  as  follows:  The  light  which  is  to  be 
analyzed  is  passed  into  the  interferometer,  the  two  mirrors  being 

*Rayleigh,  Phil.  Mag.  (5)  34,  p.  407. 


THE    VISIBILITY    CURVES  79 

near  together  and  adjusted  to  be  parallel  to  each  other.  The 
visibility  of  the  fringes,  that  is,  the  contrast  between  the  bright 
and  dark  fringes,  is  called  100.  The  screw  of  the  instrument  is 
then  turned  through  a  whole  turn  and  the  visibility  again  esti- 
mated. It  will  generally  be  less  than  100.  This  estimation  of 
visibility  requires  some  practice.  This  practice  may  be  obtained 
by  mounting  between  Nicols  a  convex  and  a  concave  quartz  lens 
of  the  same  curvature.  If  these  lenses  are  cut  parallel  to  the 
crystallographic  axis  and'  set  so  that  their  axes  are  at  right 
angles  to  each  other,  circular  fringes  similar  to  those  in  the  inter- 
ferometer will  be  seen.  As  the  lenses  are  rotated  about  the  line 
of  sight  as  an  axis,  the  visibility  varies  in  a  way  which  can  be 
calculated  from  the  angle  of  inclination  of  the  axes  of  the  lenses 
with  the  plane  of  polarization  of  the  analyzer.  For  if  a  represent 
that  angle,  then  the  two  extreme  values  Iv  and  L2  of  the  resultant 
intensity  will  be  respectively  1  and  cos2  2a,  and  therefore 

r_  /!-/«        1  ~  COS2  2a 

~ 


-  /2  ~   1  -  COS2  -2a 

Having  trained  the  eye  with  such  an  arrangement  the  visibility  is 
estimated  at  each  revolution  of  the  screw,  and  these  estimates  are 
plotted  as  ordinates,  the  corresponding  differences  of  path  being 
the  abscissae.  Even  if  the  entire  curve  is  not  worked  out,  con- 
siderable information  can  be  obtained  from  a  determination  of  the 
differences  of  path  which  correspond  to  the  minima. 

Experiment 

DETERMINE  THE  DISTRIBUTION  ix  THE  CADMIUM  LINES 
Apparatus  and  adjustments  as  in  the  previous  chapter. 
MEASUREMENTS.  —  Using  as  a  source  of  light  a  cadmium  tube, 
the  light  is  first  passed  through  an  ordinary  spectroscope  so  that 
only  one  radiation  at  a  time  passes  into  the  interferometer.     Start- 
ing near  the  position  of  the  white-light  fringes,  the  visibilities 


80  MANUAL    OF    ADVANCED    OPTICS 

which  correspond  to  the  gradually  increasing  differences  of  path 
are  observed  as  has  been  described.   It  is  well  to  observe  the  fringes 
through  a  small  telescope  focused  for  parallel  rays.     If  a  telescope 
can  not  be  used,  a  card  with  a  small  hole  in  it  should  be  mounted, 
in  front  of  the  instrument  to  insure  keeping  the  eye  at  the  same' 
point  during  the  observations.     The  observations  are  then  plotted 
as  ordinates  in  a  curve,  the  difference  of  path  being  the  abscissae. 
From  the  curve  thus  obtained  we  find  A  the  difference  in  path] 
which  corresponds  to   V  =  50.     If  the  curve  is  periodic,  corre- 
sponding to  a  double  source,  we  must  take  the  envelope  of  the 

maxima  for  making  this  calculation,  i.e.,  the  part  of  the  curve; 

x* 
represented   in   our  equation  by  2    A2.     The  half  width  of  the 

22A.       22 

source  is  then =  - — X2.     If  the   line  is  double,  the  distance 

9         A 

between  the  sources  is  determined  from  D,  the  period  of  the 
curve,  i.e.,  the  distance  between  the  maxima  or  the  minima; 

\   2 

for,  as  was  shown  above,  \1  —  X2  =  —  The  ratio  of  the  intensities 

of  the  two  lines  may  be  obtained  approximately  from  the  heights 
of  the  first  maximum  and  minimum.  Thus  the  visibility  at  the 
first  maximum  is  always  100.  If  at  the  first  minimum  it  is,  say 
20,  and  if  a  and  b  represent  the  two  intensities,  a  + 1)  -  100, 

a  —  1)  =  20,  and,  therefore,  —  =  —  =  0.7  approximately. 

(I        o 

EXAMPLES 

1.  The   red    radiation    from    cadmium  (X  =  6438  •  10~7)  was 
observed,  and  the  curve  shown  in  Fig.  23  b  obtained.     Since  this 


FIGURE  23 


THE   VISIBILITY    CURVES  81 

curve  is  iiot  periodic  we  may  conclude  that  the  line  is  single. 
The  value  of  A  is  seen  to  be  A  =  138,  hence  the  half  width  of 
the  source  is,  from  equation  (24) , 

—A2  =  .0066  -  10-7mm. 

A 

X" 

The  equation  of  the  visibility  curve  would  then  be  F=2  (138)a. 
The  curve  marked  a  shows  the  distribution  which  is  seen  to 
correspond  to  a  very  nearly  homogeneous  source. 

2.  The    green    radiation    of    cadmium    (X  =  5086  •  10"7)   was 
observed  and  the  curve  shown  in  Fig.  24  I  obtained.     Since  the 


FIGURE  24 

curve  is  periodic  with  a  single  period  it  corresponds  to  a  double 
line.     The  period  of  the  curve  is  seen  to  be  D  =  115.     Hence  the 

\    2 

distance  between  the  lines  is  Xl  —  X2  =  jr-  =  .022  -  10~7mm. 

Further,   F=  50  for  X=  120,  i.e.,  A  =  120,  therefore,  the  half 

22 

width  of  the  line  is  '—  X2  =  .0048  •  10-7mm. 

A 

The  value  of  V  at  the  first  maximum  is  100,  at  the  first  mini- 
mum 66,  hence—  =  .2  nearly.     Hence  the  equation  of  the  curve 

would  be  represented  by 

.2 


F=2    ('»)'  cos 

115 

The  corresponding  distribution  is  represented  at  the  left  of  the 
figure. 

3.  The  next  curve,  Fig.  25,  represents  the  envelope  of  the  visi- 
bility curve  for  sodium  (X  =  5890  •  10~7).     The  period  which-deter- 


82 


MANUAL   OF    ADVANCED    OPTICS 


mines  the  separation  of  the  two  lines  Dl  and  D2  has  already  been 
found  to  be  .58  mm.  (cf.  page  61).     Hence  this  period  is  omitted 


FIGURE  25 


from  the  curve.  As  it  stands,  the  curve  represents  the  distribution 
of  each  of  the  sodium  lines  upon  the  supposition  that  it  is  the  same 
for  both.  The  curve  is  seen  to  have  two  periods,  one  of  50  and  one 

X2 
of  150.  Corresponding  to  the  period  50,  we  have-y-=. 069  •  10~7mm. , 

A2 
while  for  the  period  150  we  have  -=r=  .023  •  10~7mm. 

The  value  of  A  is  seen  to  be  156,  and  so  the  half  width  of  each 
of  these  lines  is  .0063  •  10~7mm.  ' 

The  ratio  of  the  intensities  corresponding  to  the  first  period  is 
found  to  be  .7,  and  that  corresponding  to  the  second  .2,  hence  the 
equation  of  the  curve  is  represented  by 


F=2 


50 


loO 


In  connection  with  this  chapter  the  student  should  read  Michelson, 
Phil.  Mag.  (5)31,  p.  338,  1891;  Phil.  Mag.  (5)  34,  p.  280,  1892;  Journal  de 
Physique  (3)  3,  p.  5,  1894;  B.  A.  Reports,  1892,  p.  170;  Trav.  et  Mem.  du 
Bureau  Internat.  des  Poids  et  Mesures,  XI,  p.  1,  1894.  Also  some  further 
developments  of  the  same  method  are  given  by  Perot  and  Fabry,  C.  R. 
126,  pp.  34,  331,  407,  1561,  1624,  1706,  1779;  Ann.  de  Chim.  et  de  Phys.  (7) 
16,  pp.  115,  289;  C.  R.  130,  p.  653. 

Some  important  applications  of  the  method  will  be  found  as 
follows:  Michelson,  "On  the  Broadening  of  Spectral  Lines,"  Astrophys- 
ical  Journal,  2,  p.  251,  1894;  "Radiation  in  the  Magnetic  Field,"  Phil.  Mag. 
(5),  44,  P-  109,  1897;  46,  p.  348,  1898;  A  strophysical  Journal,  7,  p.  130,  1898. 


VII 


THE  PRISM  SPECTROMETER 
Theory 

To  understand  the  conditions  which  must  be  fulfilled  in  order 
that  the  spectrometer  may  be  used  with  the  greatest  efficiency,  it  is 
necessary  to  discuss  first  some  of  the  optical  properties  of  prisms. 

An  optical  prism  is  a  transparent  solid,  two  of  whose  faces  at 
least  are  plane  surfaces,  which  intersect  in  a  line.  This  line  of 
intersection  is  called  the  edge  of  the  prism.  The  angle  inclosed 
by  the  two  plane  surfaces  is  called  the  refracting  angle  of  the 
prism,  and  will  be  denoted  in  what  follows  by  A.  A  plane 
passing  through  the  prism  parallel  to  the  edge  and  perpendicular 


FIGURE  26 

to  the  plane  bisecting  the  refracting  angle  is  called  the  optical 
base,  and  a  plane  perpendicular  to  the  edge  is  called  a  principal 
plane. 

Let  CAB  (Fig.  26)  represent  the  section  of  a  prism  by  a  prin- 
cipal plane.  Let  S  represent  a  source  of  light,  and  800' R  the 
path  of  a  ray  from  that  source  through  the  prism.  Draw  NOe 

83 


84  MANUAL    OF    ADVANCED    OPTICS 

and  N'O'e  perpendicular  to  the  faces  CA  and  BA  at  the  points  0 
and  0'  respectively.  Continue  the  line  SO  to  #,  and  00'  to  $', 
and  draw  through  0',  0' H  parallel  to  SO. 

The  angle  NOS  between  the  normal  NO  and  the  direction  of 
the  incident  ray  SO  is  called  the  angle  of  incidence,  and  is 
denoted  by  i.  As  it  is  measured  from  the  normal  NO,  it  may  be 
either  positive  or  negative.  It  is  denned  as  positive  when  the 
incident  ray  SO  and  the  refracting  angle  A  lie  on  opposite  sides 
of  the  normal.  It  is,  therefore,  negative  when  the  incident  ray 
lies  between  the  normal  and  the  angle  of  the  prism. 

Similarly  the  angle  N'  0' R  is  called  the  angle  of  emergence 
and  is  denoted  by  i' .  It  is  denned  as  positive  when  the  emergent 
ray  O'R  and  the  refracting  angle  A  lie  on  opposite  sides  of  the 
normal  N'O'.  It  is,  therefore,  negative  when  the  emergent  ray 
lies  between  the  normal  and  the  refracting  angle. 

The  angles  eOO'  and  eO'O  are  called  angles  of  refraction,  and 
are  denoted  by  r  and  r'  respectively.  The  angle  r  is  positive 
when  the  *,  to  which  it  corresponds,  is  positive,  and  negative 
when  i  is  negative.  Similarly,  r  is  positive  or  negative  according 
as  the  i',  to  which  it  corresponds,  is  positive  or  negative. 

The  angle  HO' R  being  the  angle  through  which  the  ray  is 
bent  by  its  passage  through  the  prism,  is  called  the  angle  of  devi- 
ation and  is  denoted  by  8. 

From  Fig.  25  we  see  that  the  following  relations  exist: 

HO'R  =  HO'S'  +  S'O'R  =  8 

HO'S'  =gOO'  =  i-r 

S'0'R  =  i'  -r'. 

Hence  8  =  i+ i' -  (r +  r'). 

But  feO'  =  r  +  r'  =  A,  (29) 

therefore,  8  =  i  +  i'  -  A .  (30) 

The  index  of  refraction  of  one  medium  with  respect  to  another 
is  denned  as  the  ratio  of  the  velocity  of  light  in  the  one  medium 


THE    PRISM    SPECTROMETER  85 

to  that  in  the  other.     Thus,  if  V  represent  the  velocity  in  one 
medium,  V  that  in  the  other,  and  ^  the  index  of  refraction, 


As  is  well  known,  this  ratio  is  equal  to  that  of  the  sines  of  the 
angles  of  incidence  and  refraction,  that  is, 

F  j  sin  i  _  sin  i' 
V    i  sin  r      sin  r' 

We  have,  then,  as  the  fundamental  equations  of  the  prism, 


sin  I 

(*  =  —  -  •> 
sin  /• 

»  =  — 
sm  r 

r  +  r'  =  A. 


(31) 


To  obtain  the  general  equation  which  connects  the  index  of 
refraction  with  the  angles  J,  i  and  if  we  proceed  as  follows: 
From  equations  (31)  we  have 

sin  i'  =  \L  sin  r'  =  /x  sin  (A  —  r)'9 
if  we  expand  sin  (A  —  r)  and  substitute  for  sin  r  its  value  -  and 

for  cos  r  its  value  —  -v/^-sin2  «',  this  equation  reduces  to 


sin  i'  =  sin  A*/ 'f  —  sin2  i  —  cos  A  sin  i.  (32) 

This  equation  holds  in  general  without  any  conditions  imposed 
upon  the  quantities  involved.  Since,  however,  the  quantity  whose 
value  is  to  be  determined  from  measured  values  of  the  others  is 
usually  the  index  of  refraction  /A,  and  since  it  is  a  matter  of  some 
difficulty  to  measure  the  angle  of  incidence  i  with  accuracy,  it  is 
generally  advisable  to  use  the  prism  in  one  of  two  particular 
positions. 


86  MANUAL   OF   ADVANCED    OPTICS 

One  of  these  particular  positions  is  determined  •  by  the  condi- 
tion 

i'  =  0, 

that  is,  the  ray  emerges  from  the  prism  normal  to  its  second  face. 
In  this  case  sin  *'  =  0,  8  =  i  —  A,  or  i  =  8  +  A,  r'  =  0,  and  r  =  A. 
Upon  substituting  these  values  in  the  first  of  equations  (31),  it 
readily  reduces  to 

sin  (A  +  8) 


sin  A 


(33) 


Since  in  this  case  the  determining  condition  is  i'  =  0,  and 
since  we  also  have  r'  =  0  and  r  =  A,  therefore  the  fundamental 
equation  sin  i  =  /u,  sin  r  becomes,  under  these  circumstances, 
sin  i  =  /u,  sin  A  .  But  sin  i  must  be  less'  than  unity.  Therefore, 

sin  A  <  —  .  Hence,  a  prism  can  not  be  used  in  this  particular 
position  unless  its  refracting  angle  falls  within  the  limit  prescribed 
by  this  inequality,  that  is,  unless  sin  A  <  —  • 

The  other  of  these  particular  positions  is  determined  by  the 
condition 

i'  =  i. 

In  this  case  /  =  r  =  —A  and  8  =  2i  -  A,  or  /  =  —  (A  +  8),  and  on 

</  A 

substituting  these  values  in  the  first  of  equations  (31)  it  becomes 


sin      (A  +  S) 

("  =  -     V  (34) 

sin  -A 

When  the  prism  is  used  under  this  condition,  namely  i'  =  t, 
the  deviation  8  produced  by  it  is  the  smallest  which  can  be 
obtained  with  a  prism  of  given  angle  and  index  of  refraction. 
Hence  this  position  of  the  prism  is  known  as  that  of  minimum 
deviation.  That  it  is  so  may  be  proved  as  follows: 


THE    PRISM    SPECTROMETER  87 

Since  in  equation  (30)  A  is  "a  constant  for  any  given  prism, 
the  value  of  8  will  depend  on  that  of  i  -f  i'.  Therefore  8  will  be  a 
minimum  when  i  -f  i'  is.  But  an  inversion  of  equations  (31)  gives 

i  =  sin"1  /A  sin  r,  and  i'  =  sin"1  tt  sin  (A  —  r),  hence 
i  +  f  =  sin"1  /tA  sin  ;•  +  sin"1  /A  sin  (J.  —  r). 

To  find  when  this  value  of  i  +  i'  will  be  a  minimum,  differentiate 
this  equation  with  respect  to  r,  regarding  /A  as  constant.  This 
gives 

d  (i  -f-  /')  _        /A  cos  r  /A  cos  (.4  -  r) 

dr  v/l  —  /A2  sin2  r      \^1  —  /A2  sin2  (^4  —  r) 

This  becomes  equal  to  zero  when  r  =  —  A.     A  second  differentia- 

/v 

tion  with  respect  to  r  and  a  substitution  of  —A  for  r  gives 


sm  - 


Since  /x  —  sin  ^4  is*  the  sine  of  that  angle  of  incidence  which  will 

*Z 

have  a  corresponding  angle  of   refraction  —  A,  the  value  of  the 

A 

right-hand  side  of  this  equation  will  be  real  and  positive  when 
/*  >  1.  When  /M  <  1  its  value  will  be  negative,  which  means  that 
the  corresponding  value  of  i  +  i'  is  a  maximum.  But  since,  when 
/x  <  1,  the  prism  is  optically  less  dense  than  the  surrounding 
medium,  the  deviation  will,  not  be  given  by  equation  (30),  but  by 

8  =  A  —  (i  -f  i").  Hence  in  any  case,  the  condition  r  =  —  A  will 
make  8  a  minimum.  But  r  +  r'  =  A.  Hence  the  condition 
r  =  —  A  is  equivalent  to  /•  =  •/•'  or  to  i  =  f .  Therefore,  under  this 
condition  the  deviation  produced  by  the  prism  is  a  minimum. 


88  MANUAL    OF    ADVANCED    OPTICS 

There  is  a  limit  to  the    use    of   a   prism   in   this   case   also; 

for,  under  the  condition  i  =  i'  we  have  seen  that  r'  =  —A.     Since 

& 

Bint*'  <  lit  follows  that  sin—  A  <  —    If  sin  r'  >  — ,  a  thing  which 

2  p  p 

often  occurs  in  practice,  there  is  no  sin  i'  to  correspond  to  it. 
Hence  the  ray  can  not  leave  the  prism  at  the  surface  where  this 
occurs,  but  is  totally  reflected.  The  application  of  total  reflection 
to  the  determination  of  indices  of  refraction  will  be  discussed  in 
the  next  chapter. 

Consider  now  that  instead  of  a  single  ray  we  have  given  a 
narrow  beam  of  light.    As  before  let  CAB  (Fig.  27)  represent  the 


FIGURE  27 

section  of  a  prism  by  a  principal  plane,  and  S  the  projection  of  a 
narrow  source  upon  that  plane.  Suppose  that  a  narrow  beam  of 
monochromatic  light  80 A  falls  upon  the  prism  in  such  a  way 
that  one  boundary  of  the  beam  passes  through  the  edge  A  of 
the  prism.  Let  i  represent  the  angle  of  incidence  of  the  ray  SO, 
and  i'  the  corresponding  angle  of  emergence,  and  r  and  r'  the 
respective  angles  of  refraction.  Since  the  beam  is  supposed 


THE    PRISM    SPECTROMETER  89 

*>^ 

narrow,  we  may  denote  the  corresponding  angles  for  the  ray  SA 
by  i  +  di,  i'  +  di\  etc.  We  then  have  from  equations  (31) 

sin  f  =  fji  sin  (A  -  r) 
sin  i  =  /w.  sin  r, 

whence,  by  differentiation,  regarding  /*  as  constant, 

cos  i'di'  =  —  /A  cos  (A  -  r)  dr, 
cos  idi  =  fjL  cos  7Y/r. 

Eliminating  dr  and  substituting  for  A  —  r  its  valne  r'  we  have 

7.,         cos  /•'  cos  /' 

rfi  =  -   ^ —   -  tfi.  (35) 

cos  /   cos  r 

From  this  equation  it  follows,  since  the  cosine  terms  are  always 
positive,  that  when  di  is  positive,  that  is,  i  +  di>i,  di'  is  negative, 
and  therefore  i'  +  di'<i'.  Hence,  the  two  rays  O'R  and  AR ', 
when  prolonged  backwards,  will  intersect  at  some  point  $',  that 
is,  a  virtual  image  of  the  point  8  will  be  formed  at  8'  by  the 
prism. 

Equation  (35)  is  capable  of  another  interpretation.  If  8  is 
not  a -point  source,  but  has  finite  width,  di  may  be  regarded  as 
the  angular  width  of  the  source  when  viewed  from  0;  di'  then 
represents  the  width  of  the  virtual  image  when  viewed  from  0' ' . 
From  this  it  follows  that,  for  a  given  i  and  ^',  the  width  of  the 
image  is  proportional  to  the  width  of  the  source.  Also  the  width 
of  the  image  may  be  altered  by  varying  i  and  i'.  Thus  if  i  =  90  °, 
cos  i  =  0,  and  di'  =  0,  or  the  emergent  beam  is  parallel.  If 
t"  =  90°,  cos  i'  -  0,  and  dif  =  x  . 

When  the  prism  is  in  the  position  of  minimum  deviation, 
i  =  i',  r  =  /,  and,  therefore,  di  =  di',  that  is,  the  width  of  the 
image  is  equal  to  the  width  of  the  source.  Since  in  practice 
prisms  are  most  frequently  used  in  this  position,  and  since,  as 
will  be  shown  later,  considerations  of  the  purity  of  the  spectrum 
make  it  desirable  to  have  the  width  of  the  image  as  small  as 


90  MANUAL   OF    ADVANCED    OPTICS 

possible,  the  slit  in  a .  spectroscope  is  made  infinitely  narrow  by 
placing  it  at  an  infinite  distance  by  means  of  a  lens.  Hence  the 
origin  of  the  collimator. 

If  in  equation  (35)  di  =  0,  then  di'  =  0,  that  is,  a  beam  of 
monochromatic  light  which  is  parallel  before  falling  on  the  prism 
remains  so  after  its  passage  through  the  prism. 

We  have  thus  far  been  considering  only  those  rays  from  the 
source  which  lie  in  the  plane  CAB  (Fig.  27),  i.e.,  in  a  principal 
plane.  When  we  take  into  account  also  the  rays  which,  coming 
from  the  source  S,  pass  through  the  prism  in  some  other  than 
a  principal  plane,  the  development  becomes  much  more  com- 
plicated ;  especially  if  the  source  of  light  is  a  straight  bright  line 
parallel  to  the  refracting  edge  of  the  prism.  The  virtual  image 
of  such  a  line  source  will,  in  general,  be  curved,  owing  to  the 
passing  of  some  of  the  rays  through  the  prism  obliquely  to 
a  principal  plane.* 

Up  to  this  point  the  index  of  refraction  /x  has  been  regarded 
as  constant.  While  this  is  true  for  a  prism  of  any  given 
substance  so  long  as  light  of  a  definite  wave  length  only  is  con- 
sidered, it  ceases  to  be  so  when  the  light  from  the  source  contains 
more  than  one  wave  length.  We  have,  therefore,  to  consider  now 
what  further  effects  are  produced  by  a  prism  when  the  variations 
of  the  index  of  refraction  due  to  changes  of  wave  length  are  taken 
into  account,  and  to  determine  how  the  conditions  under  which 
the  prism  is  used  alter  those  effects. 

X"The  question  as  to  how  the  index  of  refraction  depends  on  the 
wave  length  has  been  the  subject  of  a  large  amount  of  investiga- 
tion. Practically  it  has  been  found  that  the  formula  first  given 
by  Cauchyf  expresses  the  relation  between  the  two  quantities  to 

*M.  A.  Bravais,  Jour,  de  Fee.  Polyt.,  18,  p.  79,  1845.  E.  Reusch,  Pogg. 
Ann.,  117,  p.  241,  1862.  A.  Cornu,  Ann.  ec.  norm.  (2)  1,  p.  255,  1872. 
G.  G.  Stokes,  Proc.  Roy.  Soc. ,  22,  p.  309,  1874.  J.  L.  Hoorweg,  Pogg. 
Ann.,  154,  p.  309,  1875. 

f  Cauchy,  Memoir e  sur  la  dispersion  de  la  lumiere,  Prague,  1836. 


THE    PRISM    SPECTROMETER  91 

a  fair  degree  of  accuracy  in  all  cases  of  normal  dispersion  within 
the  visible  spectrum.*  This  formula  is 

K    jjO 
^  =  4,+  4' 
l<        K 

in  which  34*  and  B  are  for  any  given  substance  constants  to  be 
determined  by  experiment.  It  will  be  noted  that  when 
A  =  x  ,  fj,  =  A,  i.e.,  ^represents  the  index  of  refraction  for 
infinitely  long  waves. 

In  order  to  find  the  change  in  the  angle  of  emergence  cli'  which 
results  from  a  variation  dp  in  the  index  of  refraction,  it  is  neces- 
sary to  differentiate  equations  (31),  regarding  i  and  A  only  as 
constant.  This  gives 

0  =  /A  cos  rdr  +  sin 

cos  i'di'  =  /A  cos  r'dr'  -r  sin 

dr  +  dr  =  0. 

By  elimination  of  dr  and  dr  these  reduce  to 


cos  t   cos  r 


This  equation  gives  the  relation  between  di'  and  dp  in  terms 
of  the  angles  A,  i',  and  r.  It  is  more  practical  to  have  their 
relation  in  terms  of  the  dimensions  of  the  prism  and  its  adjuncts. 
This  may  be  accomplished  for  the  most  useful  case,f  that  in 
which  the  prism  is  in  the  position  of  minimum  deviation,  as 
follows:  Since,  under  these  circumstances  we  have  i  =  f,  r  =  r'  ', 
A  =  2r,  equation  (37)  reduces  to 

sin  *2r       _       2  sin  r 

di  =  -  -  dfM  =  -   —r-  dfji.  438) 

cos  i  cos  r  cos  i 

Suppose  that  a  parallel  beam  of  light  falls  upon  the  prism 

(*  Schmidt,  Die  Brechung  des  Liclits  in  Glasern,  Leipzig,  1874. 
f  For  the  general  case  cf.  Rayleigh,  Phil.  Mag.  (5),  7,  pp.  261,  403,  477, 
1879;   9,  p.  40,  1880;    Collected  Works,  1,  p.  415.     Also  Drude,  Theory  of 
Optics,  p.  233,  Longmans,  1902. 


92 


MANUAL    OF    ADVANCED    OPTICS 


CAB  (Fig.  28)  through  a  rectangular  opening  ab,  two  of  whose 
parallel  sides  are  parallel  to  the  edge  -A  of  the  prism,  and  whose 
plane  is  perpendicular  to  the  direction  of  propagation  of  the 
beam.  Let  the  prism  be  so  placed  that  one  boundary  bA  of  the 


beam  passes  through  its  edge  A.  Let  db,  the  width  of  the  beam, 
be  represented  by  a,  and  the  distance  oo',  which  the  other  bound- 
ary of  the  beam  travels  in  the  prism,  by  t.  Draw  Ae  bisecting 
the  angle  .4,  and  of  through  o  perpendicular  to  bA.  Then,  since 
the  prism  is  in  the  position  of  minimum  deviation,  oo'  is  perpen- 


dicular to  Ae,  and  oe  =  —t. 


From  the  figure  we  get 

t 


.      1    A          06 

sin  oAe  =  sin  —A  =  —  -  =  -  —  - 
2         oA      %oA 

,.  of       a 

cos  Aof  =  cos  i  =  -^—r  -  —  -• 
oA      oA 


sm 


Substituting  in  equation  (38)  it  reduces  to 

' 


(39)- 


Hence  we  note  that  the  amount  of  separation  di'  of  two  beams 
of  light  whose  wave  lengths  are  such  that  the  difference  of  their 
indices  of  refraction  is  dp,  depends  only  on  the  width  a  of  the 


THE    PRISM    SPECTROMETER  93 

beam,  and  on  the  excess  of  transparent  substance  of  the  prism 
traversed  by  one  side  of  the  beam  over  that  traversed  by  the  other. 
One  of  the  most  important  practical  applications  of  the  dis- 
persive power  of  the  prism  is  that  of  analyzing  a  composite  beam 
into  its  components.  It  is,  therefore,  of  great  importance  to  know 
how  far  this  analysis  may  be  carried  with  any  given  prism.  Thus 
if  we  have  given  for  analysis  a  beam  of  light  containing  the  two 
different  wave  lengths  X  and  X  +  d\,  we  wish  to  know  what  sort 
of  prism  is  needed  to  accomplish  the  task,  that  is,  to  resolve  the 
beam  into  its  component  parts.  Equations  (39)  and  (1)  answer 
this  question  completely.  Equation  (39)  gives  the  angular  sepa- 
ration due  to  the  prism  of  the  two  beams,  and  if  we  suppose  a 
lens  placed  behind  the  prism  to  form  real  images  of  the  source, 
(1)  tells  whether  the  two  images  formed  by  that  lens  will  appear 
separated  or  not.  Combining  the  two  we  see  that  the  limit  of 
resolution  is  expressed  by  the  following  equation: 


,,    *  ,     x 

di  =  —  dp.=  —> 
a  a. 

or 

*  =  -r '  (4°) 

dp.  ^ 

From  this  equation  it  appears  that,  if  it  is  desired  to  analyze  or 
resolve  with  a  prism  two  beams  of  light  which  proceed  from  the 
same  source  and  whose  indices  of  refraction  are  //.  and  p.  +  dp. 
respectively,  it  will  be  necessary  to  use  a  prism  whose  thickness  £, 

as  defined  above,  is  at  least  equal  to  -j— 

It  is  often  more  convenient  to  have  this  expression  in  terms  of 
X  and  d\  instead  of  X  and  dp..  The  value  of  dp.  in  terms  of  d\ 
maybe  obtained  from  equation  (30),  and  when  substituted  in  (40) 
there  results 


94  MANUAL    OF    ADVANCED    OPTICS 

The  dispersion  of  a  prism,  denoted  by  .Z),  may  be  defined  as 
the  ratio  of  a  change  in  deviation  rfS  to  the  corresponding  change 
in  wave  length  dX,  that  is, 

d&     aft     dp 
Tx  =  TV-  '  dX 
But 

$  =  i  +  i'  —  A,   therefore,   for   constant  A 

and  i,  and  in  consideration  of  equation  (37), 

rTS  _  dif_  _       sin  A 
dp~  dp  ~  cos  i'  cos  r 
Hence 

D. 


cos  i  cos  r 


If  the  incident  beam  contains  all  possible  wave  lengths,  that 
part  of  it  which  corresponds  to  each  particular  wave  length  will, 
on  account  of  the  dispersion  of  the  prism,  form  its  own  particular 
image  of  the  source  at  the  focus  of  a  lens  suitably  placed  behind 
the  prism.  These  images  will  be  a  series  of  parallel  bright  lines 
which  overlap  and  form  a  bright  band  of  light.  Such  a  band  is 
called  a  spectrum.  The  spectrum  is  said  to  be  the  purer,  the  less 
the  successive  images  which  unite  to  form  it  overlap.  Since  each  of 
these  elementary  bands  is  produced  by  waves  whose  lengths  vary 
over  a  small  interval  dX,  we  may  use  this  change  in  wave  length 
within  the  band  as  a  convenient  measure  of  the  purity,  that  is, 
we  may  define  purity,  denoted  by  P,  as  the  reciprocal  of  dX.  Or, 
better,  if  we  choose  A.  as  the  unit  of  measure,  the  purity  may  be 
defined  as 


Let  0  be  the  angular  width  of  the  elementary  image  due  to 
wave  length  X  alone.     The  center  of  this  image  will  receive  light 

from  the  neighboring  images  whose  centers  lie  within  a  range  of  —  0 

</ 

on  either  side  of  it,  that  is,  from  those  images  included  within  a 


THE    PRISM    SPECTROMETER  95 

region  (18  =  0.  Hence,   since  the  change   in  wave  length  in  this 
interval  is  tfA,  the  illumination  in  the  center  of  each  elementary 

image  will  be  produced  by  wave  lengths  varying  from  A d\  to 

X  +  —  d\.     To  get  a  value  for  d\  divide  the  equation  d&  =  0  by  d\. 
We  have,  since  the  interval  is  small, 

d\      d\ 
But 


Therefore, 


and 


_        ^ 
d\~    0  ' 


But  from  equation  (35) 

-  cos  /'  cos  / 

^  =  f7i  =  -  ^  -  rZi. 
cos  i  cos  r 

Therefore,  in  consideration  of  equations  (42),  (43),  and  (44), 
*     cos  i'  cos  r  sin  A        dp. 


di    cos  i  cos  /•'         cos  i  cos  /•'  d\ 


When  the  source  is  narrow,  6  is  the  limit  of  resolution  of  the 
opening,  that  is,  0  =  —  ;  therefore,  in  this  case,  equation  (44) 
becomes 


Under  this  condition  —  is  called  the  resolving  power  of  the  prism. 

ft  A 

Denoting  it  by  R  we  have, 

R  =  aD.  (46) 


96  MANUAL    OF    ADVANCED    OPTICS 

From  this  it  appears  that  the  resolving  power  of  a  prism  of 
given  thickness  is  directly  proportional  to  both  the  width  of  the 
incident  beam  and  the  dispersion  of  the  prism. 

In  the  position  of  minimum  deviation  equation  (42)  reduces  to 


But  R  =  aD,  therefore,  in  consideration  of  equation  (36), 


From  this  it  appears  that,  with  a  given  width  of  beam,  the 
resolving  power  of  a  prism  is  proportional  directly  to  the  thickness 
of  the  prism,  and  inversely  to  the  cube  of  the  wave  length. 

For  further  discussion  of  the  properties  of  prisms  and  prism  spectro- 
scopes the  student  is  referred  to  Kayser,  Handbuch  der  Spectroscopie,  1, 
p.  253  seq.  and  490  seq.  ,  Leipzig,  1900,  where  a  complete  bibliography  will 
be  found.  Also  important  discussions  by  Helmholtz,  Physiologische 
Optik,  3d  Ed.,  p.  290  seq.,  Leipzig,  1896;  Rayleigh,  Phil.  Mag.  (5),  8,  pp. 
261,  403,  477,  1879;  9,  p.  40,  1880;  also  in  his  Collected  Works,  1,  p.  415 
seq.  ;  Czapski,  Winkelmann's  Handbuch  II,  1,  p.  152  seq. 


Experiments 

I.  DETERMINE  THE  INDEX  OF  REFRACTION  OF  A  PRISM 

APPARATUS. — The  spectrometer  consists  of  a  collimator  SA 
(Fig.  29)  and  a  telescope  BE,  both  mounted  at  right  angles  to 
a  vertical  axis  CD  and  movable  about  that  axis,  and  a  graduated 
circle  C,  upon  which  the  angle  between  the  telescope  and  colli- 
mator can  be  read  with  the  help  of  the  vernier  v.  Both  the 
telescope  and  the  collimator  should  be  so  mounted  as  to  be 
movable  about  horizontal  axes  aa,  so  that  they  may  be  adjusted 
to  have  their  collimation  axes  parallel  to  each  other  and  perpen- 


THE    PRISM    SPECTROMETER 


97 


dicular  to  the  axis  CD.  It  is  usually  possible  to  clamp  the 
collimator  and  the  graduated  circle  to  the  base  of  the  instrument. 
The  clamp  which  fastens  the  telescope  to  the  graduated  circle 
contains  a  tangent  screw  which  permits  a  slow  motion  of  the 
telescope.  The  center  of  the  graduated  circle  should  coincide 


3 


FlGUBE  29 


with  the  axis   CD,  and  it  is  useful  to  have  a  small  hole  bored 
through  that  center  parallel  to  that  axis. 

ADJUSTMENTS. — TJie  axes  of  the  telescope  and  collimator  must 
intersect  the  vertical  axis  of  the  instrument.  If  the  telescope  and 
collimator  are  rigidly  mounted  so  that  they  are  not  movable  about 
a  vertical  axis,  we  assume  that  this  adjustment  has  been  properly 
made  by  the  maker.  In  any  case  the  adjustment  can  be 
tested  with  sufficient  accuracy  by  sticking  a  straight  metal 
rod  into  the  hole  in  the  center  of  the  graduated  circle,  and 
observing  this  rod  through  the  slit  of  the  collimator.  To  adjust 
the  telescope  the  eyepiece  must  be  removed  and  the  rod  viewed 


98  MANUAL    OF   ADVANCED    OPTICS 

as  in  the  case  of  the  collimator.  When  the  adjustment  is  correct, 
the  rod  seems  to  cover  the  center  of  the  objective. 

Telescope  and  collimator  must  ~be  focused  for  parallel  rays. 
The  telescope  may  he  pointed  out  of  the  window  and  focused  on 
a  distant  object,  but  it  is  better  to  place  a  prism  on  the  prism 
table,  set  the  telescope  perpendicular  to  one  of  its  faces,  and 
observe  the  reflection  upon  the  prism  face  of  the  cross-hairs  in  the 
teleseope.  In  order  to  do  this  easily  the  telescope  should  be 
fitted  with  a  Gauss  eyepiece,  or  have  some  other  provision  for 
illuminating  the  cross-hairs.  When  the  cross-hairs  are  illumi- 
nated and  the  telescope  is  perpendicular  to  the  prism  face,  two 
sets  of  cross-hairs  will  be  visible  in  the  field  of  view.  The  two 
should  be  brought  nearly  into  coincidence  and  the  telescope 
so  focused  that  when  the  eye  is  moved  about  behind  the  eyepiece 
the  two  sets  show  no  parallax  with  respect  to  each  other.  Having 
thus  focused  the  telescope  the  collimator  is  easily  adjusted  for 
parallel  rays  by  observing,  in  the  telescope,  the  image  of  the  slit, 
and  focusing  the  collimator  until  that  image  shows  no  parallax 
with  respect  to  the  cross-hairs.  The  adjustment  can  also  be 
effected  by  placing  on  the  prism  table  of  the  instrument  a 
plane-parallel  plate  of  glass  in  such  a  way  that  the  light  from  the 
slit  is  reflected  to  the  telescope  at  oblique  incidence.  The 
reflected  image  of  the  slit  in  the  telescope  will  appear  double 
unless  both  telescope  and  collimator  are  focused  for  parallel 
rays. 

Telescope  and  collimator  must  be  perpendicular  to  the  axis  of 
the  instrument  and  the  surfaces  of  the  prism  must  be  parallel  to 
that  axis.  To  attain  this  the  prism  should  be  mounted  on  a 
leveling  tripod,  so  that  one  of  its  faces  AB  is  perpendicular  to 
the  line  joining  two  of  the  leveling  screws.  The  prism  so 
mounted  is  then  set  upon  the  prism  table  a  little  to  one  side  of 
the  center  of  the  graduated  circle,  so  that  when  the  telescope  and 
collimator  are  brought  into  line  and  the  prism  face  AB  is  made 


THE    PRISM    SPECTROMETER  99 

parallel  to  that  line,  part  of  the  field  of  view  of  the  telescope  is 
unobstructed,  and  it  is  possible  to  have  an  image  of  the  slit 
formed  in  the  focal  plane  of  the  eyepiece.  The  slit  should  then 
be  turned  so  that  it  is  horizontal,  and  the  telescope  and  collimator 
adjusted  by  means  of  the  screws  bb  until  the  image  of  the  slit 
falls  upon  the  intersection  of  the  cross-hairs  in  the  telescope. 
The  telescope  should  then  be  turned  about  the  axis  of  the  instru- 
ment through  any  angle,  and  the  prism  turned  through  such  an 
angle  that  the  light  from  the  slit  is  so  reflected  from  the  prism 
face  AB  that  the  image  of  the  slit  again  appears  in  the  field  of 
view  of  the  telescope.  In  general  this  image  of  the  slit  will  not 
fall  upon  the  intersection  of  the  cross-hairs.  It  should  be  made 
to  do  so  by  adjusting  the  leveling  screws  of  the  prism.  The  tele- 
scope is  then  turned  so  as  to  be  perpendicular  to  the  face  AB 
of  the  prism.  In  order  to  set  it  accurately  perpendicular  to  that 
face,  it  is  necessary  to  illuminate  the  cross-hairs  and  observe  their 
image  reflected  from  the  face  AB.  When  this  image  coincides, 
with  the  cross-hairs  themselves,  the  telescope  is  perpendicular  to 
the  face  of  the  prism.  The  cross-hairs  and  their  image  should  be 
brought  into  coincidence  by  adjusting  the  telescope.  The  tele- 
scope and  prism  should  then  be  brought  into  their  original  posi- 
tions, and  the  image  of  the  slit  observed  directly.  It  will  in 
general  no  longer  fall  upon  the  intersection  of  the  cross-hairs.  It 
should  be  made  to  do  so  by  adjusting  the  collimator  by  means  of 
the  screw  b.  This  operation  must  be  repeated  until  the  irnaga  of 
the  slit  remains  on  the  intersection  of  the  cross-hairs  in  all  three 
of  the  positions.  When  this  is  accomplished  the  telescope  and 
collimator  are  perpendicular  to  the  axis  of  the  instrument,  and 
the  prism  face  A  B  is  parallel  to  that  axis. 

It  is  then  necessary  to  make  the  other  face  of  the  prism, 
namely  AC,  parallel  to  the  axis  CD.  To  accomplish  this  it  is 
merely  necessary  to  set  that  face  perpendicular  to  the  telescope. 
It  will  be  remembered  that  the  face  AB  of  the  prism  was  set  per- 


100  MANUAL    OF    ADVANCED    OPTICS 

pendicular  to  the  line  which  joins  two  of  the  leveling  screws  of 
the  tripod  upon  which  the  prism  is  mounted.  The  adjustment  of 
the  face  A  C  should  be  made  entirely  with  the  other  screw  of  the 
tripod,  because  a  movement  of  this  third  screw  of  the  tripod  will 
not  tip  the  face  AB,  but  only  rotate  it  about  an  axis  perpendicu- 
lar to  its  plane. 

MEASUREMENTS. — It  is  first  necessary  to  measure  the  angle  A 
of  the  prism.  The  telescope  is  set  perpendicular  to  the  face  AB 
of  the  prism,  and  its  position  read  upon  the  graduated  circle.  It 
m .then  set  perpendicular  to  the  face  AC  of  the  prism,  and  its 
position  read  in  the  same  way.  The  angle  through  which  the 
telescope  turns  in  making  these  two  settings  is  the  supplement  of 
the  angle  of  the  prism.  To  measure  the  angle  8  of  deviation, 
turn  the  slit  so  that  it  is  vertical,  and  illuminate  it  with  mono- 
chromatic light, — or  better,  with  sunlight.  Place  the  telescope 
and  collimator  in  line,  bring  the  image  of  the  slit  upon  the 
cross -hair,  and  read  the  position  of  the  telescope  upon  the  gradu- 
ated circle.  The  prism  should  then  be  introduced  and  placed  in 
such  a  position  that  the  angle  of  incidence  of  the  light  from  the 
collimator  upon  it  is  nearly  equal  to  the  angle  of  emergence. 
The  adjustment  for  minimum  deviation  is  made  by  rotating  the 
prism  about  a  vertical  axis  while  observing  the  spectral  image  of 
the  slit  in  the  telescope.  As  the  prism  is  turned  that  image 
moves  in  the  field  of  view,  and  it  will  be  found  that  at  one  par- 
ticular position  of  the  prism  that  image  is  nearer  the  direct  image 
of  the  slit  than  in  any  other  position.  The  cross-hair  is  set  upon 
the  spectral  image  when  the  prism  is  in  this  position,  and  the 
angle  through  which  the  telescope  has  been  turned  read  upon  the 
graduated  circle.  If  sunlight  is  used,  the  cross-hair  is  set  upon 
the  particular  Fraunhofer  line  for  which  the  corresponding  index 
is  required.  Having  thus  measured  the  angle  of  the  prism  and 
the  angle  of  deviation,  the  index  of  refraction  is  easily  found 
with  the  help  of  equation  (34) . 


LINE 

WAVE  LENGTH 

DEVIATION    INDEX  (C 

B 

6870  - 

io-7 

47° 

34' 

24" 

1 

C 

6563  • 

io-7 

47° 

53' 

0" 

1 

D 

5890  • 

io-7 

48° 

46' 

46" 

1 

E 

5270  • 

io-7 

50° 

0' 

12" 

1 

F 

4861  • 

io-7 

51° 

14' 

46" 

1 

G 

4308  • 

io-7 

53° 

14' 

28" 

1 

THE    PRISM    SPECTROMETER  101 

EXAMPLE 

A  hollow  prism  filled  with  bisulphide  of  carbon  was  used. 
The  angle  of  the  prism  was  59°  50f  =  A.  The  following  deviations 
were  observed  for  the  lines  of  the  solar  spectrum  as  indicated: 

(CALCULATED) 

1.6060 
1.6192 
1.6285 
1.6409 
1.6532 
1.6727 
The  temperature  was  18 °C. 

II.  DETERMINE  THE  DISPERSION  CURVE 
Apparatus  and  adjustments  as  in  Experiment  I. 
MEASUREMENTS. — It  is  merely  necessary  to  measure  the  index 
of  refraction  in  the  way  described  above,  for  several  different 
wave  lengths,  and  then  to  plot  the  indices  as  ordinates  with  the 
corresponding  wave  lengths  as  abscissae.  If  it  is  desired  to  deter- 
mine the  constants  of  the  Cauchy  dispersion  equation  [equation 
(36)],  any  two  values  of  the  index  of  refraction  with  their  corre- 
sponding wave  lengths  may  be  used.  It  is  desirable  as  a  test  of 
the  accuracy  of  the  work  to  measure  the  indices  corresponding 
to  several  of  the  Fraunhofer  lines,  and  to  calculate  the  A  and  B 
of  the  Cauchy  equation  from  each  pair  of  such  indices. 

EXAMPLE 

From  the  values  of  /*,  obtained  in  the  example  of  Experiment  I, 
the  following  values  of  A  and  B  of  equation  (36)  were  obtained : 

INTERVAL  A 

B-E  1.5805 

C-F  1.5786 

D  -  G  1.5777 

mean  1.5789  =  A 

.'.  B.=  1.735  •  IO-8 


102  MANUAL    OF    ADVANCED    OPTICS 

III.  DETERMINE  THE  RESOLVING  POWER  or  THE  SPECTROMETER 
Apparatus  and  adjustments  as  in  Experiment  I. 
MEASUREMENTS.  —  Having  determined  the  B  of  equation  (36), 
it  is   in  addition  necessary  to  measure  ^,   the  thickness  of  the 
prism.     If  the  incident  beam  has  a  rectangular  cross-section,  it  is 
evident  from  Figure  28  that  the  value  Of  t  is  given  by 

2«  sin  —A  ' 

t  =  -  ;—•  (48) 

cos  ^ 

If,  however,  the  entire  aperture  of  the  collimator  is  used,  and  if 
a'  represent  the  diameter  of  that  aperture,  then,  because  the 
cross-section  of  the  beam  is  circular, 


Hence  the  equation  for  the  determination  of  t  is,  in  the  case  of 
an  incident  beam  of  circular  cross-section, 

2a'  sin  ^A 

#  =  0.82  -  :  --  (49) 

cos  i 

It  is  to  be  noted  that  the  resolving  power  of  a  prism  is  different  in 
different  parts  of  the  spectrum. 

An  interesting  check  upon  equation  (47)  is  obtained  by 
altering  «,  and  consequently  t,  until  a  given  resolution  is  obtained. 
This  is  accomplished  by  limiting  the  width  of  the  beam  by  two 
cards  held  upon  the  front  face  of  the  prism.  The  distance 
between  the  cards  should  be  varied  until  two  known  lines  are  no 
longer  resolved.  The  two  sodium  lines  are  a  convenient  pair  for 
this  purpose.  If  d  is  the  distance  between  the  cards  upon  the 
front  face  of  the  prism,  when  the  sodium  lines  cease  to  appear 
double,  then  it  is  evident  that 

t  =  26?  sin—  A. 
*  Verdet,  Legons  D'optique  Physique,  Vol.  I,  p.  301  seq. 


THE    PRISM    SPECTROMETER  103 

The  value  of  -j-  in  this  case  is  about  1000.     It  will  in  general  be 
CIA. 

found  that  the  experimentally  determined  value  of  the  resolving 
power  is  larger  than  the  theoretical  one.  This  is  natural,  as  the 
theoretical  value  is  the  limit  of  resolution  beyond  which  it  is 
impossible  to  go,  and  the  fact  that  the  observed  limit  is  somewhat 
higher  merely  goes  to  show  the  imperfection  of  our  eyes  and 
our  instruments. 

EXAMPLE 

Two  cards  were  held  over  the  objective  of  the  collimator  and 
moved  until  the  two  sodium  lines  just  ceased  to  be  resolved.  In 
this  position  their  distance  apart  was  a  =  3.5  mm.  From  equa- 
tion (48),  t  =  5.98,  since  i  =  -  (8  +  A)  =  54°  18'.  Hence,  from  equa- 

/v 

tion  (47),  using  the  value  of  B  given  on  p.  101, 

*=¥  \   x 

The  theoretical  value  of  R  for  the  two  D  lines  is  -j-  =  983.  x 

To  get  some  idea  of  the  reso\ying  power  of  the  instrument 
without  stops  in  different  parts  of  tne  spectrum,  use  was  made  of 
equation  (46)  in  connection  with  the  observations  in  Experiment  I. 
The  value  of  R  varies  over  the  intervals  chosen,  hence  the  num- 
bers obtained  are  only  averages  over  the  interval. 

INTERVAL  d&  rfx          d\  ~ 

C-D          53' 46"  =  3226"  =  .01564  radians    .0000673    232 
F-G     1°  59' 42"  =  7182"  =  .03482       "         .0000553    630 

Since  in  this  case,  if  a  represent  the  diameter  of  the  objective, 
a  =  0.82«',  and  since  a  was  40  mm.,  a  =  32.8,  therefore, 

INTERVAL         R  ( =  aZ>) 

C-D  7610 

F-  G          20700 


104  MANUAL   OF   ADVANCED    OPTICS 

The  extreme  values  of  R  for  the  lines  B  and   G  are  added. 
The  calculation  is  made  from  equations  (47)  and  (49). 

LINE  K 

B         6020 
G       24400 


VIII 

TOTAL   REFLECTION 
Theory 

In  Chapter  VII  we  have  deduced  the  general  equation  (32)  for 
the  prism.     Mention  was  also  made  of  the  fact  that  when  the 

internal  angle  of  incidence  r'  becomes  so  large   that  sin  /  =  - 

the  light  can  not  escape  from  the  prism,  but  is  totally  reflected. 
This  limiting  angle  of  total  reflection  may  then  be  used  to 
determine  /A.  At  that  angle  sin  i'  =  1,  and  hence  (32)  becomes 


1  =  sin  A  v/  'fj?  —  sin2  i  —  cos  A  sin  i. 
If  this  equation  be  solved  for  /x2  we  get 

cos  A  +  sin 


Hence,  to  determine  /*  by  this  method  it  is  necessary  to 
measure  the  refracting  angle  A  of  the  prism,  and  the  angle  i  of 
incidence, 

If  the  totally  reflecting  surface  be  covered  with  a  liquid  of  index 

it*',  then  at  the  limiting  angle  of  total  reflection  sin  i"  =  I  =  —  —  » 
and  equation  (32)  becomes  for  this  case 


/A'  =  sin  A  v//A2  -  sin2  i  —  cos  A  sin  i.  (51) 

Experiments 

I.  DETERMINE  THE  INDEX  OF  REFRACTION  BY  TOTAL  REFLECTION 
The  apparatus  and  adjustments  are  those  of  the  last  chapter. 
The  prism  used  should  be  for  convenience  a  total  reflecting  prism 
of  90°. 

105 


106  MANUAL   OF    ADVANCED    OPTICS 

MEASUREMENTS. — Place  the  prism  on  the  prism  table  of  the 
spectrometer  and  allow  diffused  light  to  fall  upon  the  face  AB  as 
shown  in  Fig.  30.  Observe  the  reflected  light  with  the  telescope 
and  find  that  position  of  the  prism  and  telescope  in  which  half  of 
the  field  of  view  is  bright  and  the  other  half  less  bright.  Set  the 
cross-hair  of  the  telescope  on  the  line  of  separation  between 
these  two  portions  of  the  field  and  read  the  position  of  the  tele- 
scope on  the  circle.  Then  set  the  telescope  perpendicular  to  the 
face  BC,  and  again  read  the  circle.  The  difference  between 
these  two  readings  will,  because  of  the  symmetry  of  the  figure,  be 


FIGURE  30 

the  angle  i,  which  corresponds  to  the  angle  i'  =  90°.  Note  that 
in  the  case  drawn  the  refracting  angle  of  the  prism  is  at  A,  and 
therefore  since  it  and  the  incident  ray  are  on  opposite  sides  of  the 
normal,  i  is  positive. 

If  it  is  desired  to  determine  the  index  /  of  a  liquid  (//</*) 
wet  the  surface  A  C  with  the  liquid.  It  will  probably  be  neces- 
sary to  press  a  film  of  the  liquid  between  the  prism  face  and  a 
piece  of  glass.  Observe  again  the  limiting  angle.  In  this  case 
the  angle  i  will  probably  be  negative.  The  index  can  then  be 
calculated  from  equation  (51). 

The  same  prism  can  be  used  to  measure  indices  of  solid  sub- 
ptances  less  dense  than  the  prism  itself  by  pressing  between  the 
surface  AC  and  the  substance  whose  index  is  to  be  determined  a 


TOTAL    REFLECTION  107 

liquid  like  oil  of  cassia,  whose  index  is  greater  than  those  of  the 
substance  and  the  prism.  Equation  (51)  holds  for  this  case  also, 
the  symbols  having  the  same  meaning  as  in  the  preceding  case. 

EXAMPLE 

Using  a  total  reflection  prism  and  sodium  light  it  was 
found  that  A  =  45°,  i  =  5°  30'.  Hence,  from 
equation  (50) /u,  =  1.5133. 

As  a  check  the  deviation  S  at  minimum  deviation  was 
also  measured,  the  result  being  8  =  25°  47'. 
Hence,  from  (34) /x  =  1.5134. 

The  surface  AC  was  then  coated  with  a  water  film, 
and  the  value  of  i  determined  as  i  =  -  25°  50'. 
Hence,  from  (51) //=  1.3329. 

Using  a  denser  total  reflection  prism  and  sodium  light 
it  was  found  that  A  =  45°,  i  =  11°  2'.  Hence;, 
as  above /x  =  1.6170. 

The  hypothenuse  of  the  prism  was  then  coated  with  a 
thin  layer  of  cassia  oil,  and  a  plate  of  the  glass 
whose  index  of  refraction  was  determined  with 
the  interferometer  (page  65),  was  placed  upon  the 
oil.  The  value  of  i  was  then  measured  as 
i  =  -  42°  57'  20".  Hence,  from  (51) V  =  1.5189. 

For  other  methods  of  determining  indices  from  the  angle  of  total 
reflection,  the  student  is  referred  to  Kohlrausch,  Wied.  Ann.  4,  p.  Jl. 
Abbe,  "Apparate  zur  Bestimmung  des  Brechungsvermogen, "  Jena,  1874. 
Pulfrich,  Zs.  fur  Instrk.  7,  p.  55,  1887;  8,  p.  47,  1888;  15,  p.  389,  1895; 
19,  p.  9,  1899.  Czapski,  Zs.  fur  Instrk.  10,  p.  346,  1890. 


IX 
THE   DIFFRACTION    GRATING 

Theory 

The  conditions  under  which  the  maxima  and  minima  of  illu- 
mination are  formed  by  the  interference  of  the  light  waves  that 
have  passed  through  two  parallel,  rectangular  openings  or  slits, 
have  been  discussed  in  Chapter  II  and  are  expressed  by  equations 
(4)  and  (5),  page  20.  If  we  now  consider,  in  the  same  way,  a 
large  number  n  of  such  equal,  equidistant,  parallel  slits,  all  of 
which  lie  in  one  plane,  we  shall  find,  in  general,  that  the  position 
of  a  maximum  will  be  determined  by  an  equation  similar  to  equa- 
tion (5)  in  which  the  2  is  replaced  by  n;  that  is,  the  condition  for 

a  maximum  will  be 

nd  sin  0  =  nm\.  (52) 

Similarly  the  condition  for  a  minimum  will  be 

nd  sin  6  =  (nm  +  l)\  (53) 

Such  a  series  of  parallel  slits  is  called  a  plane  grating.* 

In  general  the  positions  of  the  maxima  and  minima  will  be 
determined  by  these  equations.  There  are  certain  cases,  however, 
when  one  or  more  of  the  maxima  may  be  wanting.  According  to 
equation  (2)  each  individual  opening  has  a  minimum  in  a  direction 
determined  by  a  sin  0  =  m\.  But  from  equation  (52)  the  maxima 
are  determined  by  d  sin  0'  =  m'\.  When  the  0's  are  the  same, 

a       m  _.. 

T  =  — ; ••'  (°4) 

d      m 

*  Of.  Drude,  Theory  of  Optics,  p.  222  seq. ,  Longmans,  1902.  Kayser, 
Handbuch  der  Spectroscopie,  /,  p.  416  seq.,  Leipzig,  Hirzel,  1900.  Ray- 
leigh,  Phil.  Mag.  (4)  47,  p.  193,  1874.  Quincke,  Pogg.  Ann.  146,  p.  1, 1872. 
Rowland,  Phil.  Mag.  (5)  35.  Mascart,  Ann.  ec.  norm.  1,  p.  219,  1864; 
TraiU  d'optique,  1,  p.  364  seq. ,  Paris,  1889. 

108 


THE    D-IFFRACTION    ORATING 


109 


Hence,  when  this  condition  happens  to  be  fulfilled  the  correspond- 
ing maximum  is  wanting. 

We  have  thus  far  considered  that  the  incident  beam  fell 
normally  upon  the  grating.  The  introduction  of  this  condition 
has  made  the  explanation  of  the  equations  which  determine  the 
positions  of  the  maxima  and  minima  very  simple.  It  is,  however, 
immaterial  whether  the  difference  of  phase  nm  between  the 
outside  rays  of  the  beam  is  introduced  entirely  behind  the 
grating,  or  part  of  it  before  the  grating  and  the  rest  behind  it. 
Part  of  this  difference  of  phase  will  be  introduced  before  the 
grating  if  the  incident  beam  fall  upon  it  at  some  angle  other  than 
a  right  angle.  Then  let  AB  (Fig.  31)  represent  the  grating, 


'N 


A  ,- 


N 


FIGURE  31 


and  SABS'  the  incident  beam,  making  an  angle  NAS  =  i  with  the 
normal  A  JV  to  the  grating.  Let  AR  be  the  direction  correspond- 
ing to  a  maximum;  call  the  angle  which  AR  makes  with  the 
normal  AX'  to  the  grating,  the  angle  of  diffraction,  and  denote  it 
as  above  by  0.  The  difference  of  path  between  the  two  outside 
rays  AR,  BR'  is  now  represented  by  DA -{-AD'.  This  must, 
therefore,  be  substituted  for  AD  in  the  deductions  that  lead  to 


110  MANUAL   OF    ADVANCED    OPTICS 

equation  (52).  But  DA  =  nd  sin  i,  and  AD'  =  nd  sin  0,  there- 
fore, 

DA  +  AD'  =  nd  (sin  *  +  sin  6), 

and  equation  (52)  becomes  under  these  circumstances 
nmX  =  nd  (sin  •&'  +  sin  0). 

Should  the  direction  AR  fall  between  the  normal  AN',  and  the 
direct  ray  SAI,  the  difference  of  path  between  the  outside  rays 
AR'  and  BR'  would  be  given  by  AD  —  BD',  and  the  equation 

reduces  to 

nm\  =  nd  (sin  i  —  sin  6). 

Hence,  the  complete  equation  is,  after  dropping  the  n9 

m\  =  d  (sin  i  ±  sin  0),  (55) 

whereby  it  is  to  be  remembered  that  the  negative  sign  is  to  be  used 
when  the  diffracted  ray  AR  lies  on  the  same  side  of  the  normal 
as  the  direct  ray  A  I. 

The  dispersion  of  a  grating,  denoted  by  D,  may  be  denned  as 
in  the  case  of  the  prism,  as  the  ratio  of  the  change  in  the  angle 
of  diffraction  dO  to  the  corresponding  change  in  the  wave  length 

dX,  that  is,  D  =  -j-;  or,  from  equation  (52), 

llA. 

(56) 


d   cos  0 

From  this  equation  it  is  seen  that  the  dispersion  varies  directly 
with  m,_  that  is,  with  the  order  of  the  spectrum.  Hence  the 
spectrum  of  the  second  order  will  be  twice  as  long  as  that  of  the 
first  order.  Further,  the  dispersion  is  inversely  proportional  to 
the  grating  space  d.  For  this  reason  makers  of  gratings  have 
made  it  their  aim  to  make  d  as  small  as  possible.  The  dispersion 
also  varies  inversely  as  the  cos  0.  If  cos  0=1,  i.e.,  0  =  0,  the 
diffracted  ray  is  normal  to  the  grating  and  the  dispersion  is  a 
minimum.  Since  the  cosine  varies  very  little  from  the  value 
unity  when  the  arc  varies  through  a  small  angle  on  either  side  of 


THE    DIFFRACTION    GRATIXG  111 

zero,  the  dispersion  curve  is,  under  this  condition,  most  nearly  a 
straight  line.  Hence  the  spectrum  formed  under  this  condition 
is  called  a  normal  spectrum. 

The  purity  of  grating  spectra  may  be  measured  in  the  same 

"way  as  that  of  the  prismatic  spectra,  that  is,  as  before,  P  =  — » 

j  or,  when  the  source  is  infinitely  narrow,  the  resolving  power, 

X  X 

n  equals  —     The  value  of  -j-  for  gratings  may  be  determined  as 

follows:  By  equation  (52)  the  direction  of  a  maximum  for  wave 
length  X  +  dX  is,  for  normal  incidence,  determined  by 

nd  sin  6  =  nm  (X  +  <7A), 

and  by  equation  (53)  that  of  the  minimum  of  the  same  order  for 

wave  length  A  by 

nd  sin  &  =  (nm  -f  1)A. 

The  limit  of  resolution  is  reached  when  6  and  &  in  these  two 
equations  are  identical.  Hence, 

nm  (A  -f  dX)  =  (nm  +  1)A, 
or 

^ -*-«».  (57) 

Two  conclusions  can  be  drawn  from  this  equation.  First,  if 
m  is  constant,  that  is,  if  we  observe  spectra  of  the  same  order, 
,  the  resolving  power  of  different  gratings  will  be  proportional  to 
the  number  of  lines  which  they  contain. 

The  second  conclusion  may  be  reached  as  follows:  From 
equation  (52)  we  have,  for  normal  incidence, 

nd    .     - 
nm  =  —  sin  0, 
A 

but  nd  =  1  is  the  total  width  of  the  grating,  therefore, 

A       / 
-  =  —  sin  0. 
((X       A 


112  MANUAL   OF    ADVANCED    OPTICS 

If  in  this  equation  sin  6  is  constant,  that  is,  if  the  spectra  are 
observed  at  a  constant  angle  of  diffraction,  the  resolving  power  of 
all  gratings  of  the  same  width  is  the  same,  and  is  consequently 
independent  of  the  number  of  lines. 

Further,  I  cos  0  is  the  width  of  the  diffracted  beam,  denoted 
by  «',  that  is,  I  cos  9  =  a  ;  or, 


But  by  equation  (56) 


/i     a 
cos  6  =  — • 


_  w      1     _  nm    I 

d   cos  0  ~  nd   a   ' 


or,  since  nd  =  7,  and  nm  =  7?, 

D  =  ?r,     or,    R  =  a'D,  (58) 

exactly  as  in  the  case  of  the  prism.  It  is  to  be  noted,  however, 
that  a  is  the  width  of  the  diffracted  beam,  not  of  the  incident 
beam. 

Experiments 
I.     DETERMINE  THE   CONSTANT  OF  THE  GRATING 

APPARATUS. — The  spectrometer  is  used  in  these  experiments 
as  in  those  described  in  the  last  chapter.  The  grating  should  be 
so  mounted  that  it  can  be  placed  upon  the  prism  table  of  the 
instrument  and  be  adjustable  about  two  horizontal  axes,  one  of 
which  is  parallel,  and  the  other  perpendicular  to  its  surface. 

ADJUSTMENTS. — The  grating  should  be  so  set  upon  the  prism 
table  that  its  surface  lies  directly  over  the  center  of  the  graduated 
circle  of  the  instrument  and  its  lower  edge  is  parallel  to  the  prism 
table.  The  only  farther  adjustment  is  to  make  its  surface 
parallel  to  the  axis  of  the  instrument.  This  is  done,  if  the 
instrument  has  been  adjusted  as  described  in  the  last  chapter,  by 
merely  setting  the  grating  perpendicular  to  the  telescope.  If  the 


THE    DIFFRACTION    GRATING  113 

instrument  has  not  been  so  adjusted,  the  process  there  described 
must  be  used,  the  surface  of  the  grating  taking  the  place  of  the 
face  AB  of  the  prism. 

MEASUREMENTS.  —  Allow  monochromatic  light,  —  or  better, 
sunlight, — to  fall  upon  the  slit.  If  a  transmission  grating  is  to 
be  used,  it  is  better  to  set  it  normal  to  the  incident  light.  To 
attain  this,  set  the  cross-hairs  of  the  telescope  on  the  undiffracted 
image  of  the  slit.  Then  set  the  grating  perpendicular  to  the  tele- 
scope by  means  of  the  reflection  of  the  cross-hairs  in  the  usual 
way.  Read  the  position  of  the  telescope  at  this  point.  Then 
turn  the  telescope  until  the  first  spectral  image  falls  on  the  cross- 
hairs, and  again  read  its  position.  The  angle  through  which 
it  has  been  turned  in  this  operation  will  be  the  angle  0  of  diffrac- 
tion for  the  spectrum  of  the  first  order  (m  =  1).  For  the  sake  of 
a  check,  this  angle  should  be  read  on  both  sides  of  the  central 
undiffracted  image  of  the  slit.  It  is  also  well  to  determine  the 
angles  of  diffraction  for  spectra  of  several  different  orders.  The 
constant  of  the  grating  is  then  calculated  with  the  help  of  equa- 
tion (52),  the  wave  length  of  the  light  used  being  taken  from  the 
table  of  wave  lengths.  If  the  constant  of  the  grating  is  known, 
the  same  measurements  can  be  used  to  determine  wave  lengths. 
If  the  solar  spectrum  is  used,  of  course  a  given  Fraunhofer  line 
is  used  instead  of  the  spectral  image  of  the  slit. 

If  the  grating  used  is  a  reflection  grating,  it  will  probably  be 
impossible  to  observe  with  perpendicular  incidence,  because  the 
spectra  of  lower  order  fall  so  near  the  collimator  that  they 
can  not  be  observed  in  the  telescope.  In  this  case,  the  grating  is 
set  up  so  that  the  light  is  incident  at  any  angle  i.  The  telescope 
is  then  set  perpendicular  to  the  grating,  and  its  position  read 
on  the  graduate  circle.  The  angle  of  incidence  is  determined  by 
setting  the  cross-hair  upon  the  directly  reflected  image  of  the  slit, 
and  the  angles  of  diffraction  by  setting  it  on  the  spectral  images. 
The  angles  of  diffraction  are,  of  course,  all  measured  from  the 


114  MANUAL    OF    ADVANCED    OPTICS 

normal  to  the  grating,  and  their  sign  is  minus  if  they  lie  on  the 
same  side  of  the  normal  as  the  direct  reflected  image.  The 
grating  space  d  is  then  easily  obtained  from  equation  (55),  the 
wave  length  being  taken  from  the  table. 


EXAMPLE 

Using  sunlight  as  a  source,  a  plane  grating  was  adjusted  in 
the  spectrometer.  The  angle  of  incidence  was  i  =  48°  17'  30". 
The  deviations  for  the  first  three  spectra  for  three  of  the  solar 
lines  were  observed  as  follows : 

LINE  A  «j  02  03 

C  6563  •  10~7  21°  58'  20"  0°  3'  30"  21°  48'  10" 

A  5890  •  10-7  24°  19'  20"  4°  21'  10"  14°  58'  10" 

F  4861  •  10~7  28°    2'  40"  11°  10'    0"  4°  48'  10" 

Hence  from  equation  (55)  the  following  values  of  n0l=  —  )  were 
calculated : 

LINE  »0 

C  567.6 

A  568.8 

F  568.9 


Mean       568.4 

II.  FIND  THE  RESOLVING  POWER  or  THE  GRATING 
Apparatus  and  adjustments  as  in  Experiment  I. 
Having  determined   the   constant  ^,  the   resolving  power   is 

merely  m  — ,  in  which  I  is  the  length  of  the  ruling  and  m  the  order 

of  the  spectrum.  As  in  the  case  of  the  prism  the  resolution  can 
be  tested  by  finding  the  width  of  beam  a  which  is  just  sufficient 
for  the  resolution  of  a  pair  of  known  lines  like  the  two  sodium 
lines. 


THE    DIFFRACTION    GRATING  115 

EXAMPLE 

A  slit  was  placed  over  the  objective  of  the  collimator  and 
closed  until  the  two  sodium  lines  in  the  first  spectrum  were  no 
longer  resolved.  The  width  of  the  slit  was  then  found  to  be 

a  =  1.28  mm.     Hence,  the  length  of  grating  used  was :  =  1.924 

cos  ^ 

mm.,  and  the  number  of  lines  in  this  width  is  n  •  1.924  =  1094  =  R, 
the  resolving  power  according  to  equation  (57). 

7/1 

The  dispersion  was  then  calculated  from  the  equation  D  =  -^-- 

INTERVAL  dd  d\  D 

C-  D         8460"  =  .0410  radians      0.0000673       610 
D-F       13400"  =  .0664       "  0.0001029       645 

From  equation  (56)  we  get  for  Dz  in  the  first  spectrum 

. -618. 

cos  0 

The  length  of  grating,  which  was  just  sufficient  to  resolve  the 
two  sodium  lines,  was  1=  1.924  mm.  For  the  first  spectrum  we 
have  0  =  21°  58'  20".  Hence,  since  I  cos  0  =  a',  the  width  of  the 
diffracted  beam,  we  have  as  the  resolving  power,  according  ta 
equation  (58), 

R  =  a'D  =  1.784  x  613  =  1093. 

The  total  width  of  the  grating  was  40  mm.  Hence  the  resolv- 
ing power  in  the  first  spectrum  is  40  x  568.4  =  22736.  It  will  be 
noted  that  this  number  is  the  same  for  all  parts  of  the  spectrum. 

For  information  as  to  the  production  and  errors  of  gratings  cf.  Row- 
land, Phil.  Mag.  (5)  13,  p.  469;  Nature  26,  p.  211.  Rayleigh,  Phil.  Mag. 
(4)  47,  p.  81,  B.  A.  Reports,  1872;  Nature  54,  p.  332. 


THE   CONCAVE   GRATING 

Theory 

In  using  a  plane  grating,  as  in  using  a  prism,  lenses  are  neces- 
sary to  make  the  light  from  the  source  parallel  and  to  form  an 
image  for  observation.  Inasmuch  as  lenses  absorb  much  of  the 
total  energy  of  vibration  that  falls  upon  them,  it  is  highly  desira- 
ble to  avoid  their  use  if  possible.  That  this  is  possible  has  been 
beautifully  demonstrated  by  Eowland,*  and  his  concave  gratings, 
by  which  he  has  been  able  to  accomplish  this  end,  have  justly 
become  the  standard  instruments  for  spectroscopic  work. 

As  the  name  implies,  a  grating  of  this  kind  is  made  by  ruling 
fine  lines  upon  a  concave  mirror.  The  use  of  lenses  in  connec- 
tion with  the  grating  is  thus  avoided,  because  the  concave  mirror 
will  form  real  images  by  itself. 

Since  the  series  of  openings  thus  formed  are  on  the  surface  of 
a  sphere,  they  can  not  be  treated  as  a  series  of  parallel  slits.  The 
theory  of  the  concave  grating  has  been  fully  treated  by  Rowland 
and  others.  The  following  simple  deduction  of  its  leading  char- 
acteristics is  abridged  from  a  complete  discussion  of  it  by  Rungeif 

Suppose  we  have  a  source  of  monochromatic  light  at  A 
(Fig.  32),  and  a  curved  reflecting  surface  MM'  at  0.  Let  P  be 
any  point  upon  the  surface  MOM'.  An  image  of  A  will  be  formed 
at  A'  whenever  the  distance  AP  +  PA'  is  the  same  for  every 

*Am.  Jour.  Sci.  (3)26,  p.  87;  Phil.  Mag.  (5)  16,  p.  197. 
f  Winkelmann's  Handbuch,   II,   1,   p.    407.     Kayser,    Handbuch    der 
Spectroscopie,  I,  p.  452. 

116 


THE    CONCAVE    GRATING 


117 


point  P  on  the  curved  surface  MM' ,  or  whenever  the  rays 
reflected  from  every  point  P  arrive  at  A'  in  the  same  phase. 
The  first  of  these  conditions  will  clearly  be  fulfilled  if  the  reflect- 
ing surface  is  an  ellipsoid  of  revolution  having  A  and  A'  as  foci. 


FIGURE  32 


Let  us  now  conceive  a  series  of  such  ellipsoids  of  revolution  con- 
structed about  A  and  A'  as  foci  under  the  condition  that  AP  +  PA' 

shall  increase  by  —  for  each  such  successive  surface.    If  the  mirror 

4 

be  now  supposed  spherical,  its  surface  will  be  divided  by  its  inter- 
sections with  this  series  of  ellipsoids  into  a  large  number  of  zones. 
From  any  two  adjacent  zones  light  will  reach  A'  in  opposite 
phase,  because  of  the  condition  by  which  the  ellipsoids  were 
drawn.  If  the  radius  of  curvature  of  the  mirror  MM'  and  the 
distances  AP  and  PA  are  great  with  respect  to  X,  these  zones 
will  have  very  nearly  equal  width,  and  will,  therefore,  extinguish 
each  other's  effect  at  A  and  produce  there  a  minimum  of  illumi- 
nation. If,  however,  this  balance  in  the  effect  of  the  zones  is 
partly  destroyed  by  so  scratching  a  line  on  the  mirror  within  the 
region  covered  by  every  pair  of  adjacent  zones  that  some  light 
shall  arrive  from  each  pair  of  zones  in  the  same  phase,  we  shall 


118  MANUAL   OF   ADVANCED    OPTICS 

obtain  illumination  there.  It  is  to  be  noted  that  the  condition 
for  illumination  at  A'  is  that  the  light  from  every  pair  of  zones 
shall  arrive  there  in  the  same  phase.  Therefore,  we  may  have 
between  the  trains  of  waves  from  any  adjacent  pairs  of  zones  a 
difference  of  phase  of  any  number  of  whole  waves,  that  is,  of  m\. 

It  is  now  necessary  to  find  how  these  lines  must  be  drawn  to 
produce  the  desired  effect. 

Let  the  spherical  mirror  MM'  now  be  referred  to  a  set  of 
rectangular  axes,  so  that  its  vertex  is  tangent  to  the  xy  plane  at 
the  origin  0.  Call  the  radius  of  curvature  p.  The  equation  of 
the  mirror  so  placed  is 

xz  +  if  +  zz  -  %Px  =  0. 

Let  A  and  A'  lie  in  the  xy  plane,  and  let  their  coordinates  be 
represented  by  «,  #,  and  a',  #',  respectively.  Then 


or,  by  performing  the  indicated  operations  and  letting  az  +  b2  =  r*, 


Upon  substituting  in  this  equation  for  2ax  its  value  —  (x*  +  y*  +  z*) 
taken  from  above,  it  becomes 


Remembering  that  x  is  of  the  second  order  with  respect  to  y  and 
z,  and  neglecting  terms  of  the  third  order,  this  reduces  to 


I         a   I  a       1  \  ,      1 
--y  +  —  (  —  --  )#2  +  T- 
rj     2r  \r~      p  /9      »r 


The   value   of  PA  may  be   obtained  in  precisely  the  same 
manner.     It  will  be  found  to  differ  from  that  of  AP  only  in  that 


THE    COXCAVE    GRATING  119 

«',  b\  r',  appear  in  it  in  place  of  «,  J,  r,  respectively.     We  may, 
therefore,  write 


+r^- 


This  equation  reduces  to  a  much  simpler  form  for  certain  particu- 
lar positions  of  A  and  A'.  In  order  to  find  the  conditions  which 
determine  these  particular  positions  of  A  and  A',  let  us  suppose 
that  the  mirror  is  so  limited  that  the  terms  in  z2  may  be 
neglected.  The  terms  in  yz  will  vanish  also  if 

a  (a      1\       a'  (a'      1 
-        +  - 


This  will  be  true  if  r*  =  op  and  r'2  =  «'p>  that  is»  ^  ^  an(i  -4'  lie  on 
a  circle  whose  center  lies  on  the  axis  of  x  at  a  distance  |-  from  the 

origin.     When  this  simple   condition   is   fulfilled,  the   equation 
reduces  to 

AP  +  PA'  =  r  +  r' 

We  are  now  in  a  position  to  answer  the  question  as  to  how  the 
lines  must  be  drawn  on  the  mirror  to  produce  the  required  effect. 
Let  e  represent  the  difference  in  millimeters  between  the  values 
of  the  y's  which  correspond  to  the  nth  and  the  n  +  1st  lines. 
The  distance  between  the  centers  of  their  adjacent  zones  will  also 
be  e.  Since  when  A  and  A'  are  fixed,  as  we  have  supposed  them, 
r  4-  r'  is  independent  of  the  position  of  P  on  the  mirror,  the 
question  as  to  the  illumination  at  A'  depends  for  its  answer  upon 
the  value  of  the  term  containing  y.  The  condition  for  illumi- 
nation at  A'  is  that  the  light  arriving  there  from  any  zone,  the 
nth  for  instance,  shall  differ  in  phase  from  that  arriving  from 


120  MANUAL    OF    ADVANCED    OPTICS 

the  next  zone,  the  n  +  1st,  by  a  whole  number  of  wave  lengths 
m\.     This  difference  of  phase  will  accordingly  be  determined  by 


or,  since  this  must  be  equal  to  m\,  by 

*&+Si-**-  (5°) 


Hence  e,  the  difference  in  the  values  of  the  y's  for  two  consecutive 
lines,  is  constant.  That  is,  the  lines  must  be  equally  spaced 
along  a  chord  of  the  mirror  perpendicular  to  the  x  axis. 

If  conditions  are  so  arranged  that  the  image  formed  at  A'  is 
observed  only  when  A'  lies  upon  the  axis  of  #,  V  becomes  zero, 
and  equation  (59)  reduces  to 

e  -  =  m\.  (60) 

r 

But  —  =  sin  i,  where  i  is,  as  usual,  the  angle  of  incidence.  Hence, 
equation  (60)  may  be  written  in  the  form 

e  sin  i  =  m\.  (61) 

The  distance  A  A'  is,  when  b'  =  0,  equal  to  p  sin  t,  or  taking 
equation  (61)  also  into  account, 

AA'-?0±-  (62) 

6 

Hence  A  A'  is  proportional  to  the  wave  length  X. 

The  use  of  the  concave  grating  under  the  condition  V  =  0  is 
especially  to  be  recommended,  because,  when  the  point  of  observa- 
tion lies  upon  the  normal  to  the  grating,  the  dispersion  is,  as  in 
the  case  of  the  plane  grating,  very  nearly  constant  throughout  a 
considerable  range  on  either  side  of  the  normal. 

Equations  (56)  and  (57),  expressing  the  dispersion  and  resolving 
power  of  a  plane  grating,  apply  to  the  concave  grating  also. 


THE    CONCAVE    GRATING 


121 


Experiments 

I.  DETERMINE  THE  CONSTANT  OF  THE  GRATING 
APPARATUS. — As  shown  above  it  is  best  to  mount  the  concave 
grating  in  such  a  way  that  its  center  of  curvature  coincides  with 
the  point  of  observation.  This  may  be  accomplished  in  every 
position  of  the  grating  if  the  slit,  the  center  of  the  grating,  and 
the  point  of  observation  lie  upon  a  semicircle  whose  diameter 
passes  through  the  center  of  curvature  and  the  center  of  the 
grating,  and  is  equal  in  length  to  the  radius  of  curvature  of  the 
grating.  The  best  method  of  fulfilling  this  condition  is  that 

B 


0 


\  A 

FIGURE  33 

adopted  by  Rowland.  The  grating  and  the  observing  eyepiece  are 
firmly  mounted  on  the  ends  of  a  rigid  arm  GO  (Fig.  33),  whose 
length  is  equal  to  the  radius  of  curvature  of  the  grating.  This 
arm  is  supported  at  each  end  upon  a  small  carriage,  and  these 
carriages  maybe  moved  along  the  tracks  AB  and  AC,  respect- 
ively. If  these  two  tracks  are  accurately  perpendicular  to  each 
other,  and  if  the  slit  is  placed  at  J,  then  in  any  position  of  the 
arm  GO  the  points  G,  0  and  A  will  lie  on  a  semicircle,  which 
fulfills  the  required  conditions. 

ADJUSTMENTS.* — The  tracks  AB  and  AC  must  be  straight, 

*Cf.  Ames,  Phil.  Mag.  (5)27,  p.  369.     Astronomy  and  Astrophys.  11, 
p.  28,  1892. 


MANUAL   OF    ADVANCED    OPTICS 

level,  and  perpendicular  to  each  other.  These  tracks  are  usually 
mounted  upon  heavy  steel  or  wooden  beams  with  adjusting  screws 
so  that  they  can  be  raised  or  lowered  and  moved  laterally.  The 
adjustment  for  straightness  can  be  made  by  tightly  stretching 
beside  the  track  a  fine  piano  wire  or  silk  thread,  and  then  bringing 
the  track  parallel  to  this  either  with  the  adjusting  screws  or  by 
filing. 

The  tracks  may  be  made  ^horizontal  with  the  help  of  a  good 
spirit  level,  or  with  a  cathetometer  set  up  at  some  point,  as  Z>, 
and  focused  on  the  upper  edge  of  the  track. 

The  tracks  may  be  made  perpendicular  to  each  other  by  the 
3-4-5  rule,  i.e.,  by  measuring  from  their  point  of  intersection  a 
distance  of  3  units  on  one  track  and  of  4  units  on  the  other;  the 
distance  along  the  hypothenuse  between  the  two  points  thus  deter- 
mined, should  be  made  to  equal  5  units.  One  of  the  beams 
upon  which  the  track  is  fastened  is  usually  so  mounted  as  to  allow 
of  a  lateral  motion,  in  order  to  permit  this  adjustment  to  be 
made. 

The  arm  GO,  mounted  upon  its  carriages,  should  then  be  put 
in  place.  The  axes  about  which  the  arm  turns  with  respect  to 
the  carriages  should  be  directly  over  the  center  of  the  track,  and 
be  marked  at  the  top  by  a  small  hole  or  point.  It  is  assumed 
that  the  maker  of  the  carriages  has  attended  to  this.  The 
grating  should  then  be  set  in  place.  The  center  of  its  surface 
should  be  tangent  to  the  axis  of  the  carriage  over  which  it  stands. 
The  grating  holder  should  be  adjustable  in  every  direction,  i.e., 
about  each  of  the  three  rectangular  axes  of  Figure  31. 

The  center  of  curvature  of  the  grating  should  fall  upon  the 
axis  about  which  the  arm  GO  turns  at  the  end  0.  To  accomplish 
this,  a  Gauss  eyepiece  is  mounted  at  0,  so  that  its  cross-hairs  are 
directly  over  the  hole  or  point  which  marks  that  axis.  The 
centers  of  the  eyepiece  and  the  grating  should  lie  in  the  same 
horizontal  plane,  i.e.,  the  distance  of  both  from  the  top  of  the 


THE    CONCAVE    GRATIXG  123 

track  should  be  the  same.  The  adjustment  then  divides  itself 
into  two  parts :  First,  the  normal  erected  at  the  center  of  the 
grating  must  intersect  the  axis  at  the  observing  end  of  the  arm, 
and  must  be  parallel  to  the  plane  of  the  tracks ;  and,  second,  the 
cross-hairs  of  the  eyepiece  and  their  image  formed  by  the  grating 
must  coincide,  i.e.,  the  cross-hairs  must  lie  at  the  center  of  curv- 
ature of  the  grating.  To  fulfil  the  first  of  these  conditions,  the 
grating  should  first  be  set  by  eye,  so  that  the  lines  upon  it  are 
approximately  vertical.  It  should  then  be  turned  about  the  hori- 
zontal and  vertical  axes,  which  are  parallel  to  the  plane  tangent  to 
it  at  its  center,  i.e.,  about  the  y-  and  the  z-axes  of  Figure  31, 
until  the  image  of  the  cross-hairs  of  the  eyepiece  nearly  coincide 
laterally  and  vertically  with  the  cross-hairs  themselves.  The  arm 
GO  must  then  be  adjusted  in  length  until  the  cross-hairs  and 
their  reflected  image  show  no  parallax.  The  arm  GO  should 
have  been  constructed  to  have  very  nearly  the  correct  length.  It 
should,  however,  have  a  splice  in  it,  which  allows  an  adjustment 
of  a  centimeter  or  so  in  length. 

The  slit  should  then  be  mounted  over  the  intersection  of  the 
tracks,  with  its  center  at  the  same  distance  from  the  tracks  as  the 
center  of  the  grating.  If  the  other  adjustments  have  been  cor- 
rectly made,  the  slit  should  be  in  focus  in  the  eyepiece  at  0.  In 
case  it  is  not  in  focus  it  should  be  moved  slightly  forward  or 
backward  in  the  direction  AB.  If  it  is  found  necessary  to 
move  it  more  than  a  millimeter  or  two  in  order  to  bring  it  into 
focus,  it  indicates  that  some  of  the  other  adjnstments  are  inac- 
curate, and  it  is  advisable  to  verify  them. 

To  be  sure  that  the  lines  of  the  grating  are  perpendicular  to 
the  plane  of  the  tracks  move  the  arm  GO  back  and  forth.  The 
spectra  will  pass  the  point  0.  The  lines  are  vertical  when  the 
spectra  remain  at  the  same  height  above  the  track  at  0  as  the 
arm  is  moved. 

The  slit  should  be  parallel  to  the  lines  of  the  grating.     This 


124  MANUAL    OF    ADVANCED    OPTICS 

adjustment  is  best  made  with  the  help  of  the  solar  spectrum. 
The  slit  should  be  illuminated  with  sunlight,  and  the  grating  set 
in  such  a  position  that  a  group  of  fine  lines  appears  in  the 
eyepiece.  While  observing  these  lines  the  slit  should  be  rotated 
about  a  horizontal  axis.  When  the  lines  are  most  sharply  defined, 
the  slit  is  parallel  to  the  lines  of  the  grating.  This  adjustment 
may  be  made  easier  by  introducing  a  narrow  opaque  object,  like  a 
medium-sized  wire,  horizontally  across  the  slit.  The  images  of 
the  slit  in  the  eyepiece  will  then  be  divided  into  two  parts,  and  will 
appear  to  run  into  points  on  the  two  sides  of  the  division.  When 
these  points  are  vertically  opposite  each  other  the  slit  is  parallel 
to  the  lines  of  the  grating. 

MEASUREMENTS. — Illuminate  the  slit  with  monochromatic 
light  of  known  wave  length,  or  better  with  sunlight, — and  move 
the  arm  GO  until  the  spectral  image  of  the  slit,  or  a  known 
Fraunhofer  line,  falls  upon  the  cross-hairs  of  the  eyepiece. 
Measure  the  distance  AO  and  the  radius  of  curvature  OG  of 
the  grating.  Now  the  angle  i  of  incidence  is  AOG,  and  the  angle 
of  diffraction  is  equal  to  zero,  hence  from  equation  (61) 

OA 

m\  =  e  sin  i  =  e  -777^' 

(J(jr 

from  which  the  value  of  e  is  readily  determined. 

The  concave  grating  has  been  used  to  make  absolute  deter- 
minations of  wave  lengths.  In  these  measurements  the  grating 
space  was  determined  by  dividing  the  width  of  the  ruling  by  the 
total  number  of  lines  upon  it.  The  results  obtained  by  different 
observers  by  this  method  differ  among  themselves  by  one  part  in 
15,000.*  The  grating  furnishes,  however,  the  most  accurate 

*  The  following  are  the  most  important  absolute  determinations  of  the 
wave  length  Dt.  5895.81,  Angstrom,  corrected  by  Thalen,  Nov.  Act. 
Upsal.  (3)  12,  p.  1.  5896.25,  Miiller  and  Kempf,  Publ.  Potsdam  Obs.  5. 
5895.90,  Kurlbaum,  Wied.  Ann.  33,  p.  159.  5896.20,  Bell,  Am.  Jour.  Sci. 
(3)  S3,  p.  167;  35.  p.  265;  Phil.  Mag.  (5)  23,  p.  265;  25,  p.  255. 


THE    CONCAVE    GRATING  125 

means  of  determining  relative  wave  lengths  after  it  has  been 
calibrated  by  known  waves.  Thus  Rowland's  relative  determina- 
tions, made  by  this  method,  have  justly  become  the  standard 
work  on  the  subject.*  For  fixing  the  absolute  lengths  he  uses  a 
mean  of  the  absolute  determinations  of  the  lengths  of  Dl9  as 
given  in  the  footnote  below.  Probably  the  most  accurate  abso- 
lute determination  of  wave  length  is  that  of  the  three  cadmium 
lines,  which  was  made  by  Michelsonf  by  the  interferometer 
method.  As  a  discussion  of  the  details  of  the  methods  employed 
to  obtain  accuracy  in  the  determinations  with  the  grating  would 
lead  us  beyond  the  scope  of  this  book,  the  student  is  referred  for 
further  information  to  the  articles  mentioned  in  the  footnotes 
or  to  Kayser's  Handbuch  der  Spectroscopie,  Vol.  1,  p.  715  seq., 
where  the  methods  are  presented  at  length. 

EXAMPLE 

To  determine  the  constant  of  the  grating  the  following 
measurements  were  made  upon  some  of  the  Fraunhofer  lines  of 
the  solar  spectrum : 

LINE 
B 
D, 
E 
F 

The  radius  of  curvature  of  the  grating  was  640.5  cm.  Hence, 
from  equation  (60)  the  following  values  of  w0(=  — )  are  obtained: 


AO  (m  =  1) 

AO  (m  =  2) 

AO  (m  =  3) 

259.8  cm. 

519.5  cm. 

779.2  cm. 

222.7    " 

445.4    " 

668.1    " 

199.2   " 

398.5    " 

597.9    " 

183.8    " 

367.6    " 

551.3    " 

LINE 

n0 

B 

590.32 

D, 

590.31 

E 

590.30 

F 

590.30 

mean 

590.31 

*  Am.  Jour.  Sci.   (3)&?,  p.  182;    Phil.  Mag.  (5)  23,  p.  257;  27,  p.  479;   36, 
p.  49. 

t  Mem.  du  Bur.  internal,  des  Poids  et  Mes.  11,  p.  1 ;  C.  R.  11G,  p.  790. 


126  MANUAL   OF    ADVANCED    OPTICS 

The  width  of  the  ruling  was  I  =  145  mm.  Hence,  the  total 
number  of  lines  upon  the  grating  is  n  =  In0  =  85,595 ;  and  therefore, 
from  equation  (57),  the  resolving  power  in  the  first  spectrum  is 
R  =  85,595.  Since  6  =  0  at  0,  the  dispersion  D  is  determined  from 

tTl 

equation  (56)  as  D  =  —  =  mnQ  =  590.31  for  the  first  spectrum.    The 

6 

width  a  of  the  beam  is  in  this  case  the  width  of  the  ruling,  i.e., 
a  =  145  mm.     Hence,  by  equation  (58),  R  =  aD  =  85,595. 


XI 

F  POLARIZED   LIGHT 

Quantitative  experiments  in  polarized  light  presuppose  some 
general  knowledge  of  the  phenomena  of  polarization.  Since  expe- 
rience has  shown  that  few  possess  this  knowledge  with  sufficient 
definiteness,  the  following  simple  exercises  are  suggested,  and  the 
student  is  advised  to  take  any  of  the  standard  texts  as  a  guide  and 
to  work  them  out  before  attempting  the  experiments  in  the  next 
four  chapters. 

1.  Make  a  dot  on  a  piece  of  white  paper  and  observe  it  perpen- 
dicularly through  a  crystal  of  Iceland  spar.     The  line  connecting 
the  two  dots  seen  is  always  parallel  to  the  line  connecting  which 
angles  of  a  perfect  rhomb? 

2.  Rotate  the  crystal  and  note  the  effect  produced.     Call  the 
ray  which  behaves  extraordinarily  the  extraordinary  ray  E,  and 
the  other  the  ordinary  ray  0. 

3.  Toward  which  angle  of  the  rhomb  is  E  bent? 

4.  Prove  that  the  two  emergent  rays  are  parallel. 

5.  Observe   the  dot   through   two   crystals   similarly  placed; 
explain  the  effect. 

6.  Observe  the  dot  through  two  crystals  oppositely  placed; 
explain  the  effect. 

7.  Place  the  crystals  so  that  their  axes  inclose  an  angle  of  -45° 
and  explain  the  effect.     Call  the  four  rays  00,  Oe,  Eo  and  Ee^  and 
state  how  you  distinguish  each. 

8.  Which  of  the  four  disappear  when  the  axes  of  the  crystals 
include  an  angle  of  90°?     Which  when  the  axes  include  an  angle 

of  180°?     Since  the  position  of  the  upper  crystal  when  Ee  disap- 

127 


128  MANUAL    OF    ADVANCED    OPTICS 

pears  differs  by  90°  from  its  position  when  Eo  disappears,  these 
two  rays  are  said  to  be  polarized  in  planes  at  right  angles  to  each 
other. 

9.  Observe  a  dot  obliquely  through  a  rhomb  of  Iceland  spar, 
and  determine,  by  means  of  the  bending  of  0  and  E,  which  travels 
the  faster  in  the  crystal. 

10.  Explain  the  construction  of  the  Nicol  prism. 

11.  Assume  that  0  vibrates  in  the  plane  perpendicular  to  the 
principal  plane  of  the  wave,  i.e.,  to  the  plane  defined  by  the  wave 
normal  and  the  optic  axis.     By  means  of  this  assumption  deter- 
mine the  plane  of  transmission  of  the  Nicol,  i.e.,  the  plane  in 
which  the  transmitted  vibrations  take  place. 

12.  Using  the  Mcol  to  determine  the  plane  of  vibration,  find 
in  which  plane  the  vibrations  of  the  light  reflected  from  a  glass 
surface  at  the  polarizing  angle  take  place. 

13.  Determine  the  plane  of  maximum  vibration  of  the  light 
transmitted  by  a  plate  of  glass  when  the  light  is  incident  at  the 
polarizing  angle. 

14.  Increase   the    number   of  plates    and    explain   the   effect 
produced  upon  both  the  reflected  and  the  transmitted  light. 

15.  Prove,  experimentally,  that  a  Wollaston  prism  makes  the 
beams  0  and  E  divergent,  and  explain  the  reason. 

16.  Observe  a  sodium  burner  through  two  Nicols.     Shut  off 
the  light  by  crossing  the  Nicols.    Insert  a  thin  crystal  and  explain 
the  effect. 

17.  Rotate  the  crystal  and  locate  its  axes. 

18.  Rotate  the  analyzer  and  explain  the  effects. 

19.  Replace  the  sodium  burner  with  a  source  of  white  light. 
Rotate  the  crystal  and  explain  the  effects.     Rotate  the  analyzer 
and  explain  the  effects. 

20.  Try  the  same  experiment  without  the  polarizer. 

21.  Replace  the  analyzer  with  a  Wollaston  prism  and  prove 
that  the  colors  of  the  two  images  are  complementary. 


POLARIZED    LIGHT  129 

22.  Replace   the   crystal   by   one   thinner  and    then   by   one 
thicker  and  explain  the  effects. 

23.  Examine  and  explain  the  effects  produced  by  passing  con- 
vergent or  divergent  plane  polarized  light  through  a  crystal  cut 
perpendicular  to  the  axis. 


XII 

KOTATION   OF   THE   PLAXE   OF   POLAKIZATIOX 

Theory 

It  has  long  been  known  that  if  plane  polarized  light  be  passed 
through  certain  substances,  the  plane  of  polarization  is  rotated 
through  an  angle  which  is  peculiar  to  each  such  substance.  On 
this  account  such  substances  are  called  optically  active.  They 
may  conveniently  be  divided  into  three  classes  as  follows : 

First,  those  substances  which  rotate  the  plane  of  polarization 
only  when  they  are  in  crystal  form  and  lose  their  optical  activity 
when  they  are  melted  or  brought  into  solution.  The  most 
important  of  these  substances  is  quartz,  which  crystallizes  in  the 
hexagonal  crystal  system.  Since  the  optical  activity  of  these  sub- 
stances is  lost  when  their  crystal  form  is  destroyed,  this  peculiar 
property  of  theirs  must  depend  only  on  the  geometrical  arrange- 
ment of  the  molecules  in  the  crystal,  and  hence  its  investigation 
falls  properly  within  the  domain  of  physics. 

Second,  those  substances  which  show  this  optical  activity  not 
only  in  the  crystal  form  but  also  when  melted  or  in  solution.  In 
this  class  belong  several  of  the  camphors  and  tartrates. 

Third,  those  substances  which  are  optically  active  only  when 
in  the  liquid  form  or  in  solution.  This  class  contains  carbon 
compounds  only.  Since  the  members  of  the  second  and  third 
classes  retain  their  optical  activity  not  only  in  solution  but  also  in 
the  vapor  form,  it  follows  that  their  power  of  rotating  the  plane  of 
polarization  depends  on  the  arrangement  of  the  atoms  in  the 
molecule,  and,  therefore,  its  study  belongs  properly  in  the 

domain  of  chemistry. 

130 


ROTATION    OF   THE    PLANE    OP    POLARIZATION  131 

For  substances  of  the  first  class,  the  amount  of  rotation  varies 
with  the  thickness  of  the  substance  traversed  by  the  light,  with 
the  wave  length  of  the  light,  and  with  the  temperature. 

It  has  been  experimentally  proved  that  the  angle,  denoted  by 
a,  through  which  the  plane  of  polarization  is  rotated  by  an  active 
substance  is  proportional  to  the  thickness  traversed.*  If  / 
denotes  this  thickness,  then 

a  =  kl, 

'  in  which  k  denotes  the  rotation  produced  by  unit  thickness.  For 
quartz,  the  values  of  k  for  a  plate  1  mm.  thick  cut  perpendicular 
to  the  optic  axis  are,  for  light  of  the  wave  lengths  corresponding 
to  the  Fraunhofer  lines  of  the  solar  spectrum,! 

FRAUNHOFER  LINE  B  C-  D  E  F  G 

k  15.75°     17.31°     21.72°    27.54°     32.76°     42.59° 

The  dependence  of  the  angle  of  rotation  upon  the  wave  length 
may  be  expressed  by  J 

A       B 

a  =TI  +TT' 
\2         A* 

in  which  A  and  B  are  constants  to  be  determined  for  each 
substance  by  experiment.  From  the  values  of  X  given  above  for 
quartz,  A  and  B  are  found  to  have  the  following  values,  when  A 
is  expressed  in  mm., 

A  =  7.1083  •  10-6         B  =  0.1477  •  10~12. 

The  variation  of  a  due  to  changes  of  temperature  is,  for  sub- 
stances of  the  first  class,  small.  For  quartz, § 

ar  =  a»  [1  +  0.000147(0]. 

*Biot,  Mem.  de  1'Acad.  ~?,  pp.  41,  91,  1817;  Ann.  chim.  phys.  (2)  10, 
p.  63. 

t  Soret  and  Sarasin,  C.  R.  95,  p.  635,  1882. 

^Boltzmann,  Pogg.  Ann.  Jubelbd.,  p.  128,  1874.  For  another  disper- 
sion equation  cf.  Lotnmel,  Wied.  Ann.  14,  p.  523,  1881. 

§Gumlich,  Wiss.  Abh.  d.  Phys.-techn.  Reichsanstalt,  2,  p.  230,  1895. 


132  MANUAL    OF    ADVANCED    OPTICS 

In  considering  the  optical  activity  of  substances  of  the  second 
and  third  classes  in  solution,  the  idea  of  specific  rotation,  intro- 
duced by  Biot,  is  found  convenient.  The  specific  rotation  of  an 
active  substance  in  solution,  denoted  by  [a],  is  defined  as  the  angle 
through  which  the  plane  of  polarization  is  rotated  by  a  column  of 
the  solution  1  dm.  long,  which  has  in  1  cc.  of  its  volume  1  gm.  of 
the  active  substance.  Thus  if  the  solution  be  obtained  by  placing 
c  gm.  of  the  active  substance  in  a  flask  and  filling  the  flask  till  it 
contain  100  gm.  of  solution,  then  by  definition, 


Since,  as  has  been  observed  above,  the  optical  activity  of  sub- 
stances of  the  second  and  third  classes  is  due  to  their  molecular 
structure,  it  has  been  found  useful  to  introduce  the  molecular 
weight  into  the  definition.  Hence,  the  molecular  rotation  is 
defined  as  the  specific  rotation  multiplied  by  the  molecular  weight 
and  divided  by  100.  The  100.  is  introduced  merely  to  avoid  large 
numbers.  Thus,  if  [M]  represent  the  molecular  rotation,  and  M 
the  molecular  weight, 


that  is,  the  molecular  rotation  is  the  angle  through  which  the 
plane  of  polarization  would  be  rotated  by  a  column  of  the  solution 
of  the  length  of  1  mm.,  which  contained  in  every  cc.  of  its  vol- 
ume 1  gram  molecule  of  the  active  substance. 

It  was  at  first  supposed  that  specific  rotation  was  a  constant 
characteristic  of  a  substance.  Later  investigation  has,  however, 
brought  out  the  fact  that  it  varies  with  the  concentration,  with 
the  nature  of  the  solvent,  and  with  the  temperature.  These  vari- 
ations are  generally  small.  Their  cause  has  been  shown  to  lie, 
for  some  substances  at  least,  in  the  incomplete  dissociation  of  the 
molecules  in  the  solution  into  their  ions.  As  a  discussion  of  this 


ROTATION    OF    THE    PLANE    OF    POLARIZATION  133 

subject  lies  beyond  the  scope  of  this  work  the  reader  who  wishes 
to  pursue  the  matter  further  is  referred  to  Landolt,  Optische 
Dreltungscermogen  organischer  Substanzen,  2d  edition,  Braunsch- 
weig, 1898. 

The  case  of  sugar,  however,  on  account  of  its  peculiar  impor- 
tance from  a  practical  point  of  view,  merits  special  attention.  The 
value  of  [a]  for  sugar,  as  determined  from  solutions  whose  concen- 
tration varies  from  5  to  30%,  is  essentially  constant.  From  a 
large  number  of  different  measurements,  its  value  at  20  °C.  for 
sodium  light  has  been  determined  as 


Its  variation  with  the  temperature  is,  according  to  the  most 
recent  work,  for  values  of  t  between  12°  and  25  °C,* 

[a]^[a]"-0.0217  (t-'20).  (64) 

Sugar  possesses  very  nearly  the  same  dispersive  power  as 
quartz.  The  following  figures  give  the  rotation  corresponding  to 
some  of  the  Fraunhofer  lines  as  measured  in  solutions  whose  con- 
centrations varied  from  10  to  20%,  the  column  of  the  liquid  used 
being  of  such  length  that  the  sodium  light  is  rotated  the  same 
amount  as  by  a  plate  of  quartz  1  mm.  thick.  f 

FRAUNHOFER  LINE  B  C  D      »          E  F  G 

Rotation  15.20°     17.23°     21.71°     27.64°     33.08°    43.14° 

A  comparison  of  these  values  with  those  given  above  for  quartz 
shows  how  nearly  the  dispersive  powers  of  the  two  substances 
correspond.  It  is  because  of  this  fact  that,  as  is  done  in  some 
forms  of  sacchari  meter,  it  is  correct  to  make  the  measurements  by 
compensating  the  rotation  produced  by  the  sugar  solution  by  a 
rotation  in  the  opposite  sense  produced  by  quartz. 

*Sch6nrock,  Zs.  f.  Instrk.  20,  p.  97,  1900. 
t  Stefan,  Wien.  Ber.  52,  II,  p.  486. 


134  MANUAL    OF    ADVANCED    OPTICS 

The  specific  rotation  of  invert  sugar  varies  much  more  than 
that  of  cane  sugar  with  changes  of  concentration  and  tempera- 
ture. Its  value  and  its  variations  with  the  concentration  and  the 
temperature  are,*  for  values  of  c  up  to  35  and  of  t  between  0° 
and  30°C., 


M;=  [a]2  -0.304(^-20),  (65) 

in  which  c'  =  —  —  c,  the  fraction  being  the  ratio  of  the  molecular 


weights  of  the  two   sugars.     The   rotation  is  in  this  case  left- 
handed. 

Cane  sugar  may  be  converted  into  invert  sugar  by  adding  to 
100  cc.  of  the  sugar  solution,  10  cc.  of  strong  hydrochloric  acid, 
and  keeping  the  mixture  at  a  temperature  of  70°  C.  for  ten  minutes. 
It  has,  however,  been  shownf  that  the  value  of  [a']^  determined 
from  solutions  thus  converted  is  not  constant,  but  depends  some- 
what on  the  relative  amounts  of  sugar  and  acid  in  the  solution. 
The  following  is  recommended  :  To  every  100  parts  of  sugar  add 
1  part  of  oxalic  acid  and  let  the  mixture  stand  for  some  hours  at 
a  temperature  of  50°  to  60°C. 

Experiment 
DETERMINE  THE  PURITY  OP  A  SAMPLE  OF  SUGAR 

APPARATUS.  —  The  essential  parts  of  the  polarimeter  are  two 
Nicols,  A  and  B  (Fig.  34),  mounted  on  the  ends  of  a  firm  hori- 
zontal bar  about  30  cm.  long.  It  should  be  possible  to  rotate  the 
analyzing  Nicol  B  about  a  horizontal  axis,  and  to  read  the  angle 
through  which  it  has  been  turned  upon  a  vertical  graduated  circle 
(7,  which  is  fastened  to  the  base  of  the  instrument.  The  accuracy 

*  Landolt,  Optische  Drehungsvermogen,  Braunschweig,  1898,  p.  526. 
t  Gubbe,  Ber.  I.  deutschen  Chem.  Ges.  18,  p.  2210. 


ROTATION    OF    THE    PLANE    OF    POLARIZATION 


135 


of  the  readings  which  can  be  made  with  this  simplest  form  of 
instrument  is  not  very  great.  The  methods  employed  in  increas- 
ing the  attainable  accuracy  of  setting  are  different  in  different 
forms  of  instrument.  The  device  most  frequently  used  is  that  of 
introducing  directly  behind  the  polarizer  a  second  doubly  refract- 

C 


B 


FIGURE  34 

ing  object,  which  has  the  effect  of  dividing  the  field  of  view  into 
two  or  more  parts  whose  planes  of  polarization  make  a  small  angle 
with  each  other.  Thus  in  the  Laurent  polarimeter  there  is  intro- 
duced behind  the  polarizer  a  thin  plane-parallel  plate  of  quartz  so 
cut  as  to  effect  a  slight  rotation  of  the  plane  of  polarization  of  the 
polarizer,  and  so  placed  as  to  cover  half  of  the  field  of  view.  In 


136 


MANUAL    OF    ADVANCED    OPTICS 


the  most  recent  form  of  instrument,  this  alteration  of  the  azimuth 
of  the  plane  of  polarization  is  effected  by  a  small  Nicol,  which 
covers  half  of  the  field  of  view,  and  whose  principal  plane  may  be 
set  to  make  a  small  angle  with  that*of  the  polarizer.  The  accu- 
racy of  setting  is  increased  by  introducing  this  small  angle 
between  the  planes  of  polarization  of  the  two  halves  of 
the  field  of  view,  because  it  is  then  impossible  so  to 
set  the  analyzer  as  to  extinguish  the  light  from  both 
halves  of  the  field  of  view  at  the  same  time.  When  the 
principal  plane  of  the  analyzer  is  perpendicular  to  the 
plane  which  bisects  the  angle  between  the  planes  of 
polarization  of  the  two  halves  of  the  field  of  view,  this 
field  appears  uniformly  illuminated,  i.e.,  the  line 
between  the  two  halves  disappears.  It  has  been  found 
that  the  eye  can  judge  more  accurately  the  position  of 
equal  illumination  of  the  two  halves  of  the  field,  than 
it  can  the  position  of  total  extinction  of  the  whole  field. 
In  order  to  make  the  line  of  division  between  the 
two  halves  of  the  field  sharp,  a  small  telescope  is  usually 
added  to  the  observing  end  of  the  instrument.  This 
telescope  is  focused  upon  the  edge  of  the  quartz  plate 
of  the  Nicol  which  has  been  introduced.  The  arrangement  of 
the  optical  parts  of  one  of  the  more  sensitive  forms  of  the  instru- 
ment is  shown  in  Fig.  35. 

In  most  instruments  the  polarizer  can  be  turned  through  a 
small  angle,  thus  allowing  an  alteration  in  the  angle  between  the 
planes  of  polarization  of  the  two  halves  of  the  field  of  view. 

ADJUSTMENTS. — If  the  instrument  has  been  properly  con- 
structed, the  only  adjustment  needed  is  that  of  the  angle  between 
the  planes  of  polarization  of  the  two  halves  of  the  field  of  view. 
Up  to  a  certain  limit,  the  smaller  this  angle  the  more  accurate  the 
setting.  It  should  be  made  as  small  as  the  brightness  of  the 
source  and  the  absorption  of  the  solution  to  be  examined  will 


FIGURE  35 


ROTATION    OF   THE    PLANE    OF    POLARIZATION  137 

permit.  The  student  should  experiment  and  set  the  angle  at  the 
point  at  which  he  can  set  with  greatest  accuracy  in  a  given 
case. 

MEASUREMENTS. — It  is  first  necessary  to  determine  the  zero  of 
the  instrument.  Using  a  sodium  burner  as  a  source  of  light,  the 
empty  tube  which  is  to  contain  the  solution  to  be  tested  is  placed 
between  the  polarizer  and  the  analyzer,  and  the  analyzer  is  rotated 
until  the  field  of  view  is  uniformly  dark.  The  position  of  the 
analyzer  is  then  read  on  the  graduated  circle.  The  analyzer 
should  then  be  rotated  through  180°,  and  the  position  again  read. 
These  two  readings  are  to  be  used  as  the  zero  readings  of  the 
instrument.  The  tube  is  then  filled  with  the  solution  to  be  tested 
and  again  placed  between  the  polarizer  and  the  analyzer.  The 
analyzer  should  then  be  rotated  until  the  field  of  view  appears 
uniformly  dark,  and  its  position  read  upon  the  graduated  circle. 
The  reading  must  be  taken  also  after  the  analyzer  has  been  turned 
through  180°.  The  mean  of  the  differences  between  these  two 
last  readings  and  the  corresponding  zero  readings  is  the  angle 
through  which  the  plane  of  polarization  has  been  rotated  by  the 
solution. 

If  the  rotation  is  large,  it  may  be  doubtful  whether  the  rotation 
is  to  the  right  or  to  the  left.  This  question  may  be  settled  either 
by  observing  with  two  different  sources  of  light  and  remembering 
that  the  rotation  for  longer  waves  is  ordinarily  less  than  for  short 
waves,  or  by  making  a  second  set  of  observations  with  a  more 
dilute  solution,  which  would*,  of  course,  give  a  smaller  rotation. 

The  polarimeter  is  most  frequently  used  in  analyzing  solutions 
of  cane  sugar.  In  this  case  the  method  of  procedure  is  as  follows : 
Add  10  to  15  gm.  of  the  sugar  to  be  tested  to  about  85  cc.  of 
distilled  water.  After  the  sugar  is  dissolved  add  water  till  the 
solution  weighs  100  gm.  Take  half  of  this  solution,  i.e.,  50  cc., 
and  add  to  it  5  cc.  of  strong  hydrochloric  acid.  Warm  this 
mixture  to  a  temperature  of  70 °C.,  and  let  it  stand  at  that  tern- 


138  MANUAL   OF   ADVANCED    OPTICS 

perature  for  ten  minutes.  The  sugar  will  then  be  converted  into 
invert  sugar.  Before  using  it  should  be  allowed  to  cool  to  the 
temperature  of  the  room. 

Two  tubes  are  usually  supplied  with  the  instrument,  one 
20  cm. ,  and  the  other  22  cm.  long.  The  shorter  tube  should  be 
filled  with  the  sugar  solution,  and  the  longer  with  the  solution  of 
the  invert  sugar.  The  rotations  which  the  two  produce  are 
then  determined,  the  sugar  rotating  to  the  right,  the  invert 
sugar  to  the  left.  The  temperature  should  be  noted,  especially 
during  the  observations  with  the  invert  sugar. 

To  reduce  the  observations,  let  a  represent  the  measured  rota- 
tion of  the  sugar,  a!  that  of  the  invert  sugar,  and  t'  the  tempera- 
tures at  which  the  respective  observations  were  made.  Then 
from  equations  (63)  and  (64), 

.=  {[«£-O.Oai7(*-ao)j^  +  /S/'   ;.,ii:      1 

in  which  ft  represents  any  rotation  which  may  be  produced  by 
substances  other  than  sugar  in  the  solution.  Similarly  from 
equations  (63)  and  (65), 

a'=  j  [«']"- 0.304  (*'-20) 

in  which  ft  is  negative,  because  active  substances  other  than  sugar 
are  not  altered  by  the  hydrochloric  acid.  The  sum  of  these  two 
equations  is,  in  consideration  of  equations  (64)  and  (65), 

,       (  r  _,      360  r  ,,,'   }     le 

ft  -f-  tt    ==     A     I  ft         -\- ft  r  ' 

(  L  ]D     342  L    J*  )  100 

from  which  the  value  of  c,  which  is  the  number  of  grams  of  sugar 
in  100  gm.  of  solution,  may  be  readily  calculated. 

EXAMPLE 

13.29  gm.  of  cane  sugar,  which  had  been  crystallized  from  a 
sugar  solution,  was  carefully  dried  over  sulphuric  acid  in  vacuo 


ROTATION    OF    THE    PLANE    OF    POLARIZATION  139 

and  dissolved  in  water,  so  that  100  gm.  of  solution  were  obtained. 
Half  of  this  was  converted  into  invert  sugar  as  above  directed. 
The  following  observations  were  made,  /  being  for  the  sugar  solu- 
tion 20  cm.,  and  for  the  invert  sugar  solution  22  cm.  : 

a  =  17°.  35,  a'  =  5°.  30,  t  =  t'  =  27°C. 
The  reduction  is  then  as  follows  : 

[a']*  =  19.657  +  .0361c'  =  20.162 
.304    #'-20    =    2.128 


=  18.034 


[a]'D  =  66.352 
M'  4-  36°  Fa'!''      85  335 

I  Q.  I       "T"  —      ""Ift     I         =  O  O  •  O  O  O 

.-.  a  +  a'  =  0.85335fc. 
But,  from  the  observations,  a  +  a'  =  22°. 65  and  I  =  2  dm. 

.-.  22.65  -1.70670 
c  =  13.27  gm. 


XIII 
ELLIPTICALLY    POLARIZED    LIGHT 

Theory 

Suppose  we  have  given  a  beam  of  plane  polarized  light  whose 
vibrations  take  place  in  the  plane  OC  (Fig.  36).  Let  00  (=  A) 
represent  the  amplitude  of  the  vibration.  Suppose,  further,  that 
a  plane-parallel  plate  of  a  doubly  refracting  crystal,  cut  parallel  to 
the  optic  axis,  be  introduced  into  the  path  of  the  plane  polarized 
beam  in  such  a  way  that  its  faces  are  perpendicular  to  the  beam, 


0 


FIGURE 


while  its  principal  planes  have  the  directions  OX  and  0  Y.  The 
incident  vibration  OC  will  be  divided  by  the  plate  into  two  com- 
ponents OF  (=  a)  and  OG  (=  &),  which  will  be  equal  respectively 
to  OCcosO  and  00  sin  0  if  0  represents  the  angle  COP.  These 
two  component  vibrations  travel  through  the  plate  with  different 
velocities;  hence,  when  they  emerge,  though  they  will  still  be 
vibrating  parallel  respectively  to  OF  and  OG,  one  will  be  in 

advance   of   the  other  by  an   amount  which  depends  upon  the 

140 


ELLIPTICALLY    POLARIZED    LIGHT  141 

thickness  of  the  crystalline  plate  and  the  difference  of  velocities 
of  the  two  components  in  it.  If  then  A  cos  %-*  —  represent  the 

original  vibration,  and  8  the  difference  in  the  optical  paths  of  the 
two  components  when  they  emerge  from  the  crystal,  tha  vibra- 
tions of  the  two  components  will  be  represented  by 


, 

X  =  a  COS  £TT  — 


f--  +  _l 


(6(5) 


respectively.  After  passage  through  the  plate  these  vibrations 
unite  into  a  single  one.  The  path  which  any  vibrating  particle 
pursues  may  be  determined  by  eliminating  t  from  equations  (66). 

The  result  is 

^ 

cos  '4ir  — 

=sin827r^  (67) 


a*       b~  ab  A. 

This  is  the  equation  of  an  ellipse  inscribed  in  the  rectangle 
whose  sides  are  Ha  and  *2b  respectively.  The  position  of  the  ellipse 
in  the  rectangle  depends  manifestly  upon  the  value  of  8.  Two 

cases  are  of  especial  interest:    First,  when  8  =  0,   or—-     In  this 

8  8 

.  case  sin  HIT—  =  0,  cos  2?r—  =  +  1,  and  the  path  of  the  particle  is 

A  A 

given  by 


i.e.,  the  emergent  light  is  again  plane  polarized,  the  direction  of 
vibration  being  parallel  to  00. 

Second,  when  8  =  —  »  -  —   In  this  case,  sin  2ir—  =  +  1,  cos  2ir—  =  0, 
•i       -±  A  A 

and  the  equation  for  the  path  reduces  to 

x* 


142  MANUAL    OF    ADVANCED    OPTICS 

i.e.,  the  path  of  the  vibration  of  the  emergent  light  is  an  ellipse 
whose   principal   axes  coincide   in   direction  with  the   principal 

planes  of  the  crystalline  plate.     When  8  =  — ,  the  motion  about  the 

ellipse  is  in  a  direction  contrary  to  that  of  the  hands  of  a  watch 

3 

(levogyre),  while  when  8  =  —A.,  the  rotation  is  in  the  same  direction 

as  that  of  the  hands  of  a  watch  (dextrogyre).     If  in  addition  it 
should  happen  that  a  =  #,  the  equation  becomes 

,  *•  +  *•- ««; 

i.e.,  the  path  of  the  vibration  is  a  circle. 

Conversely  it  is  true  that  if  any  elliptical  vibration  is  separated 
into  components  along  the  principal  axes  of  the  ellipse  the  differ- 

,  .     TT         STT 

ence  in  phase  of  those  components  is  —  or  — • 

#  A 

These  two  special  cases  of  elliptically  polarized  light  are  of 
importance,  because  they  furnish  a  method  of  finding  the  posi- 
tion of  the  axes  of  the  elliptical  vibration,  and  of  measuring  the 
relative  intensity  of  the  components  along  those  axes.  The 
method  consists  in  reducing  the  difference  in  phase  between  the 

two   components   by  — ,  and  measuring  the  angle  which  the  result- 
4 

ing  plane  polarized  vibration  makes  with  the  axes  of  the  ellipse. 
The  method  is  explained  in  detail  in  Avhat  follows . 


Experiment 

ANALYZE  AN  ELLIPTICAL  VIBRATION 

APPARATUS.  —  The  most  convenient  instrument  for  deter- 
mining the  nature  of  elliptically  polarized  light  is  an  ordi- 
nary spectrometer  to  which  has  been  added  a  pair  of  Nicols  and 
three  graduated  circles.  The  circles  CCC  (Fig.  37)  should  be 
mounted  on  short  tubes  bbl>  with  their  planes  perpendicular  to 


ELLIPTIC  ALLY    POLARIZED    LIGHT 


143 


the  length  of  the  tubes.  Two  of  these  tubes  should  be  made  to 
fit  over  the  telescope  and  collimator  at  their  objective  ends,  while 
the  third  fits  over  the  eye  end  of  the  telescope.  Into  the  end  of 
each  of  these  tubes  fits  a  collar  ddd  which  may  be  rotated  freely 


FIGURE  37 


within  the  tube,  and  to  which  may  be  fitted  either  the  Nicol 
prisms  or  the  Babinet  compensator  or  the  quarter-wave  plate  as 
desired.  These  collars  also  carry  indices  Hi  by  which  their  posi- 
tions may  be  read  on  the  graduated  circles  upon  the  tubes. 

The  Babinet  compensator  consists  of  a  pair  of  quartz  wedges 
having  the  same  angle  between  their  "faces.  One  wedge  is  so  cut 
that  the  optic  axis  of  the  quartz  is  parallel  to  the  edge  of  the 
wedge,  while  in  the  other  the  optic  axis  is  perpendicular  to  the 
edge  and  parallel  to  one  of  the  faces  of  the  wedge.  These  wedges 
are  mounted  in  a  brass  case,  the  wedge  A  (Fig.  38)  being  fastened 
to  one  side  of  the  case,  while  the  wedge  B  can  be  moved  back  and 


144 


MANUAL    OF    ADVANCED    OPTICS 


forth  by  the  micrometer  screw  C  in  a  direction  perpendicular  to 
its  edge.  The  head  of  this  screw  is  graduated,  and  the  mounting 
of  this  wedge  B  carries  a  scale  by  which  its  position  can  be 
determined. 

Suppose  light  polarized  in  a  plane  which  is  normal  to  the 
plane  defined  by  the  two  optic  axes  of  the  wedges,  and  which 
bisects  the  angle  between  those  axes,  to  fall  perpendicularly  upon 
the  wedges  as  indicated  by  the  line  NN  (Fig.  38).  On  enter- 


FlGUBE 


ing  the  wedge  A,  the  vibration  will  be  separated  into  two  equal 
components,  one  parallel  to  the  edge  of  the  wedge,  and  the  other 
perpendicular  to  it.  These  components  will  travel  with  different 
velocities  within  the  wedge,  and  will,  therefore,  emerge  with  a 
difference  of  phase,  the  magnitude  of  which  will  depend  upon  the 
color  of  the  light  used  and  the  thickness  of  the  wedge  at  the 
place  where  the  light  has  passed  through.  On  entering  the 
wedge  B,  the  light  again  separates  into  two  component  vibrations 
which  travel  with  different  velocities,  but  because  the  optic  axis 
of  this  wedge  is  perpendicular  to  that  of  the  other  wedge,  the 
component  vibration  which  traveled  slower  in  wedge  A  will  travel 
faster  in  wedge  B,  and  that  which  traveled  faster  in  A  will  travel 
slower  in  B.  It  is  evident,  then,  that  along  that  line  parallel  to 
the  edges  of  the  wedges  where  the  light  has  traveled  through 
equal  thicknesses  of  the  two  wedges,  it  will  emerge  in  exactly  the 
same  condition  in  which  it  entered,  while  along  other  lines  its 


ELLIPTICALLY    POLARIZED    LIGHT  145 

condition  will  depend  on  the  difference  of  the  thicknesses  through 
which  it  has  passed. 

If  such  a  pair  of  wedges  be  brought  between  crossed  Nicols 
and  so  oriented  that  the  optic  axes  make  angles  of  45°  with  the 
planes  of  transmission  of  the  Xicols,  the  line  along  which  the 
thicknesses  of  the  two  wedges  is  the  same  will  appear  as"a  dark 
line,  because  along  it  the  condition  of  polarization  of  the  incident 
light  is  not  altered  by  the  compensator.  If  the  combination  is 
illuminated  with  white  light,  colored  bands  will  be  seen  on  either 
side  of  this  dark  band,  because  wherever  the  difference  in  the 
thicknesses  is  such  that  a  difference  of  phase  of  a  whole  wave  for 
any  particular  color  is  introduced  by  the  compensator,  that  color 
will  be  wanting  in  the  light  transmitted  by  the  analyzer.  If  the 
combination  be  illuminated  with  monochromatic  light,  a  series  of 
equidistant  dark  bands  will  appear,  the  distance  between  the  suc- 
cessive bands  showing  how  far  it  is  necessary  to  move  along  the 
wedges  in  order  to  introduce  a  difference  of  phase  of  a  whole 
wave  of  the  particular  light  used. 

If  the  wedge  B  be  moved  by  the  screw  (7,  the  bands  will  move, 
and  the  constant  of  the  instrument  is  merely  the  number  of  turns 
which  must  be  given  the  screw  C  in  order  to  make  the  dark  bands 
move  up  one,  i.e.,  the  number  of  turns  which  correspond  to  a 
difference  of  phase  of  a  whole  wave.  By  turning  the  screw  C 
through  a  fraction  of  the  number  of  turns  which  correspond  to  a 
difference  of  phase  of  a  whole  wave,  a  difference  of  phase  of  that 
same  fraction  of  a  wave  is  introduced  along  the  line  formerly 
marked  by  the  central  dark  band.  Thus  the  Babinei  compen- 
sator allows  us  to  introduce  any  difference  of  phase  which  may  be 
desired.  It  may  also  be  used  to  measure  a  difference  which  has 
been  introduced  by  other  means,  for  such  a  difference  of  phase 
will  shift  the  central  band,  and  may  be  measured  by  counting  the 
number  of  turns  of  the  screw  (7,  which  are  needed  to  cause  the 
bands  to  return  to  their  original  position.  This  number,  divided 


146  MANUAL    OF    ADVANCED    OPTICS 

I 

by  the  constant  of  the  instrument,  will  be  the  required  difference 
of  phase.     It  is  because  of  its  use  in  this  way  to  compensate  a  • 
phase   difference  which   has  already  been  introduced,  that  thei 
instrument  has  received  its  name. 

The  quarter-wave  plate  is  a  plate  of  crystal  so  cut  and  of  such 
thickness  that  incident  plane  polarized  light  is  separated  by  the 
plate  into  two  components  at  right  angles  to  each  other,  and  one 
of  these  components  is  retarded  over  the  other  by  a  quarter-wave 
of  the  particular  light  used. 

ADJUSTMENTS. — The  tubes  carrying  the  graduated  circles 
having  been  set  in  place  upon  the  collimator  and  telescope  of  ] 
the  spectrometer,  preferably  with  the  zero  of  the  circle  upward, 
the  Nicols  must  be  mounted  in  these  tubes  so  that  their  planes  of 
transmission  are  either  vertical  or  horizontal  when  the  circles  read 
zero.  To  attain  this,  place  in  the  tube  at  the  end  of  the  collima- 
tor a  double-image  prism.  On  looking  into  the  eyepiece,  two 
images  of  the  slit  will  be  seen.  Rotate  the  double-image  prism 
until  these  two  images  are  superimposed  in  the  center  of  the 
field,  one  projecting  above  the  area  in  which  they  are  superim- 
posed, the  other  below  it.  If  the  slit  of  the  collimator  is  vertical, 
then,  when  the  images  overlap,  as  described,  the  vibrations  of  the 
light  in  one  of  the  images  will  be  vertical,  while  those  in  the  other 
will  be  horizontal.  A  Nicol,  suitably  mounted  to  fit  the  tube, 
should  then  be  placed  in  front  of  the  objective  of  the  telescope, 
and  turned  until  it  entirely  extinguishes  one  of  the  images  of  the 
slit.  The  double-image  prism  should  then  be  replaced  by  another 
Nicol,  the  index  of  the  circle  on  the  collimator  should  be  set  to 
zero,  and  the  Nicol  rotated  in  the  collar  which  holds  it  till  the 
image  of  the  slit  is  extinguished.  The  plane  of  transmission  of 
the  Nicol  on  the  collimator  is  then  either  vertical  or  horizontal 
when  the  index  by  which  its  position  is  read  stands  at  zero.  If 
the  plane  along  which  the  two  halves  of  the  Nicol  are  cemented 
together  is  vertical,  the  transmitted  vibrations  are  horizontal,  etc. 


ELLIPTICALLY    POLARIZED    LIGHT  147 

This  adjustment  can  also  be  made  by  setting  the  analyzer  so  as 
to  extinguish  the  light  reflected  from  a  vertical  plate  of  glass  at 
the  angle  of  complete  polarization.  When  this  takes  place  the 
plane  of  transmission  of  the  analyzer  is  horizontal. 

The  Nicol  in  front  of  the  objective  of  the  telescope  should 
then  be  removed  to  the  eye  end  of  the  telescope,  and  a  quarter- 
wave  plate  mounted  in  front  of  the  objective.  It  is  very  con- 
venient to  place  the  Nicol  at  the  eye  end  between  the  lenses  of 
the  eyepiece,  though  of  course  it  can  be  mounted  either  in  front 
of  or  behind  the  eyepiece.  This  Nicol  is  then  crossed  with  the 
Nicol  on  the  collimator  (the  polarizer),  and  the  quarter-wave  plate 
turned  until  the  field  is  dark.  The  planes  of  transmission  of  the 
quarter-wave  plate  are  then  respectively  vertical  and  horizontal. 
The  readings  of  the  circles  which  carry  the  quarter-wave  plate 
and  the  Nicol  on  the  eye  end  of  the  telescope  (the  analyzer) 
should  then  be  taken.  They  are  the  zero  positions  of  these  two 
parts  of  the  instrument. 

If  the  Babinet  compensator  is  to  be  used,  it  should  be  mounted 
at  the  eye  end  of  the  telescope  directly  in  front  of  the  eyepiece. 
A  cross-hair  should  be  fastened  to  the  compensator  to  mark  the 
position  of  the  central  black  fringe,  and  the  eyepiece  focused  on 
this  cross-hair.  The  constant  of  the  Babinet  must  then  be  deter- 
mined for  the  wave  length  to  be  used.  To  accomplish  this,  the 
plane  of  vibration  of  the  polarizer  should  be  turned  to  make  an 
angle  of  45°  with  the  horizontal,  and  the  compensator  set  so  that 
its  optic  axes  are  respectively  vertical  and  horizontal.  The 
analyzer  should  then  be  crossed  with  the  polarizer.  Open  the  slit 
wide  and  illuminate  it  with  monochromatic  light  of  the  desired 
color  and  count  the  number  of  turns  of  the  micrometer  screw  on 
the  compensator  which  are  necessary  to  make  the  dark  bands  in 
the  field  of  view  move  up  one. 

In  all  of  these  experiments  with  polarized  light,  a  bright 
source  should,  if  possible,  be  used.  A  section  of  a  solar  spectrum, 


148 


MANUAL    OP    ADVANCED    OPTICS 


or  filtered  sunlight,  will,  in  most  of  these  cases,  be  sufficiently 
uniform. 

MEASUREMENTS.  —  Having  adjusted  the  instrument  as  has 
been  described,  introduce  a  thin  piece  of  mica  behind  the  polar- 
izer. Let  OH  (Fig.  3y)  represent  the  direction  and  amplitude 


E 
FIGURE  39 


of  the  plane  polarized  light,  which  is  incident  upon  the  mica,  and 
OA  and  OB  the  planes  of  transmission  of  the  mica.  Let  EEE 
represent  the  resulting  elliptical  vibration  which  it  is  desired  to 
analyze.  First  consider  the  case  in  which  the  quarter-wave  plate 
is  to  be  used.  It  is  evident  that  the  elliptical  vibration  will  be 
reduced  to  a  plane  one  if  the  directions  of  the  planes  of  vibration 
of  the  quarter-wave  plate  coincide  with  those  of  the  major  axes  of 
the  ellipse,  i.e.,  with  the  directions  OE  and  OF  in  the  figure.  The 
plane  of  vibration  of  the  restiltant  plane  polarized  light  will  evi- 
dently be  represented  by  0(7,  and  hence  in  order  to  extinguish  the 
light,  the  analyzer  must  be  turned  till  its  plane  of  transmission  is 
perpendicular  to  0(7,  i.e.,  till  it  has  the  direction  OB'.  When, 
therefore,  the  quarter-wave  plate  and  the  analyzer  have  been  set 
so  that  the  light  is  totally  extinguished,  the  planes  of  vibration  of 
the  plate  will  indicate  the  positions  of  the  principal  axes  of  the 
elliptical  vibration,  and  the  tangent  of  the  angle  between  one  of 


ELLIPTICALLY    POLARIZED    LIGHT  149 

those  planes  and  the  plane  of  transmission  of  the  analyzer,  i.e., 
angle  R 'OF  (=  angle  COE),  will  determine  the  ratio  of  those  axes, 
for 


(68) 


If  the  compensator  is  used  instead  of  the  quarter-wave  plate,  it 
is  merely  necessary  to  set  it  so  that  it  will  introduce  a  difference 
in  phase  of  a  quarter-wave  and  rotate  it  until  the  central  black 
band  returns  to  the  zero  position.  The  analyzer  must  then  be 
rotated  until  the  central  black  fringe  is  blackest.  The  directions 
of  the  optic  axes  of  the  compensator  show  the  directions  of  the 
principal  axes  of  the  ellipse,  and  the  tangent  of  the  angle  between 
one  of  those  axes,  and  the  plane  of  transmission  of  the  analyzer  is 
the  ratio  of  those  axes. 

EXAMPLE 

The  analyzer,  quarter-wave  plate  and  polarizer  were  adjusted 
as  described  above.  Filtered  sunlight  was  used.  After  the  intro- 
duction of  the  mica  it  was  necessary  to  turn  the  quarter-wave 
plate  through  an  angle  of  13°  20'  and  the  analyzer  through  an 
angle  of  43°  30'  to  extinguish  the  light.  Hence,  the  principal 
axes  of  the  elliptical  vibration  form  made  an  angle  of  13°  20'  with 
the  horizontal  and  vertical  planes  respectively,  and  the  ratio  of 
their  amplitudes  was  tan  (43°  30'  -  13°  20')  =  tan  30°  10'  =  0.581. 


XIV 

THE     REFLECTION     OF     POLARIZED     LIGHT     FROM 
HOMOGENEOUS   TRANSPARENT   SUBSTANCES 

Theory 

If  a  beam  of  light  of  amplitude  A  ,  plane  polarized  in  the  plane 
of  incidence,  fall  upon  the  surface  of  a  homogeneous  transparent 
substance,  the  amplitude  A'  of  the  reflected  light  will,  according 
to  Fresnel,  *  be  given  by 


sm  (i  +  r) 


in  which  i  and  r  denote,  as  usual,  the  angles  of  incidence  and 
refraction  respectively.  When  the  index  of  refraction  of  the 
substance  is  greater  than  unity,  i.e.,  when  /«.>!,  i>r.  Hence, 
in  this  case  the  reflected  amplitude  has  a  sign  opposite  to  that  of 
the  incident  amplitude;  i.e.,  there  is  a  loss  of  half  a  wave  at 
reflection,  and  in  absolute  value  A'<A,  becoming  equal  to  A  only 
when  i  =  90°.  At  normal  incidence,  i  =  0,  and  the  right-hand 
side  of  the  equation  becomes  indeterminate.  The  limiting  value 
of  the  expression  may  then  be  found  with  the  help  of  the  con- 
sideration that,  for  small  values  of  i  and  r,  the  sine  may  be 
replaced  by  the  angle.  Thus  i  =  nr  and 


If  the  incident  light  has  the  amplitude  B,  and  is  polarized  in 

*  Fresnel,  Oeuvres,  I,  p.  640,  Paris,  1866.     Drude,  Theory  of  Optics,  p. 
282,  Longmans,  1902. 

150 


THE    REFLECTION    OF    POLARIZED    LIGHT  151 

a  plane  perpendicular  to  the  plane  of  incidence,  Fresnel's  expres- 
sion for  the  amplitude  of  the  reflected  light  is 


tan  (i  +  r) 

When  fi  >1,  i.e.,  />r,  and  when  also  i  +  r<90°,  B'  is  negative, 
i.e.,  there  is  a  loss  of  half  a  wave  at  reflection.  When  i  +  r  =  90°, 
B'  =  0.  When  i  +  r>90°,  the  light  is  reflected  without  the  loss 
of  half  a  wave.  For  the  particular  case  «"  -t-  r  =  90°  we  have 

sin  r  =  cos  /,  and  hence,  since  -.  —  =  u, 

sin  r 

tan  i  =  /n.  (72) 

The  angle  determined  by  this  equation  is  called  the  angle  of  total 
polarization.  This  equation,  first  experimentally  established  by 
Brewster,  is  known  as  Brewster  's  law.* 

If  the  incident  light  is  polarized  neither  in  the  plane  of  inci- 
dence nor  in  a  plane  at  right  angles  to  it,  but  in  a  plane  which 
makes  an  angle  0  with  the  plane  of  incidence,  then,  since  we  have 
adopted  Fresnel's  assumption  that  the  direction  of  vibration  is 
perpendicular  to  the  plane  of  polarization,  the  component  y  of 
the  amplitude  in  the  plane  of  incidence  will  be 

y  =  A  sin  0, 

while  the  component  x  perpendicular  to  that  plane  will  be 
* 

x  =  A  cos  0, 

the  re-axis  lying  in  the  surface,  the  y-axis  in  the  plane  of  incidence, 
both  perpendicular  to  the  intersection  of  these  two  planes.  The 
values  of  these  components  after  reflection  will,  therefore,  be 

sin  (i-r) 


x  =  —  x 


y  -- 


sin  (i  +  r) 
tan  (i  —  r) 


tan 


*  Brewster,  Phil.  Trans.  1815,  p,  125.  Jamin,  Ann.  de  chim.  et  phys. 
(3)  29,  pp.  31  and  263;  (3)  30,  p.  257.  Conroy,  Proc.  Roy.  Soc.  31,  p.  487, 
1881.  Rayleigh,  Phil.  Mag.  (5)  33,  p.  1,  1892;  Collected  .Works,  3,  p.  49a 


152  MANUAL   OF   ADVANCED    OPTICS 

These  two  components  will  unite  to  form  a  plane  polarized 
beam  whose  inclination  0'  to  the  plane  of  incidence  is  given  by 

tan  6'  =  £; 

*£/ 

or,  substituting  the  values  of  x'  and  y'  taken  from  above, 

.,      y  cos  (i  +  r)      cos  (i  +  r)  , 

tan  0  =  -  -  —j-. (-  = }-. f  tan  6.  (73) 

x  cos  (i  —  r)      cos  (i  —  r) 

When  the  light  is  incident  normally,  i  =  0  and  tan  0  =  tan  0'. 
As  i  increases,  0'  becomes  less  than  0  until  i  reaches  the  angle  of 
complete  polarization,  when  &  =  0.  If  i  be  further  increased, 
&  becomes  negative,  reaching  the  value  —  6  when  i-  90°. 

These  reflection  equations  of  Fresnel  have  been  frequently 
verified  experimentally.*  This  is  most  easily  done  with  the  help 
of  equation  (73).  It  is  merely  necessary  to  allow  light  which  is 
plane  polarized  in  a  plane  whose  azimuth  with  the  plane  of  inci- 
dence is  known,  to  be  reflected  from  the  surface  of  a  transparent 
medium  whose  index  of  refraction  is  known,  and  to  measure  the 
angle  of  incidence  i,  and  the  azimuth  #',  of  the  plane  of  polariza- 
tion of  the  reflected  light. 

Experiments 
I.     VERIFY  BREWSTER'S  LAW 

The  apparatus  and  adjustments  are  those  described  in  the  pre- 
ceding chapter. 

MEASUREMENTS. — A  plate  of  crown  glass  is  set  upon  the  prism 
table  of  the  spectrometer,  and  adjusted  so  that  its  faces  are  par- 
allel to  the  axis  of  the  instrument,  as  described  in  Chapter  VII, 
page  98.  Filtered  sunlight  is  allowed  to  pass  through  the  colli- 

*Rood,  Am.  Jour.  Sci.  (2)  49;  50,  1870.  Rayleigh,  Nature  35,  p.  64, 
1886;  Proc.  Roy.  Soc.  41,  p.  275,  1886;  Collected  Works  II,  p.  522.  Con- 
roy,  Proc.  Roy.  Soc.  35,  p.  26,  1883;  37,  p.  38,  1884;  45,  p.  101,  1888;  Phil. 
Trans.  ISO,  p.  245,  1889. 


THE    REFLECTION    OF   POLARIZED    LIGHT  153 

mator  and  fall  upon  one  of  the  faces  of  the  glass  plate,  and  the 
reflected  light  is  received  in  the  telescope,  having  the  analyzer  in 
front  of  the  objective.  At  the  angle  of  complete  polarization  all 
of  the  vibrations  of  the  light  reflected  from  the  glass  take  place  in 
a  vertical  plane.  The  analyzer  should,  therefore,  be  set  so  that 
its  plane  of  transmission  is  horizontal,  and  the  telescope  and  glass 
plate  revolved  about  the  axis  of  the  instrument  until  the  position 
is  found  in  which  the  analyzer  extinguishes  all  of  the  reflected 
light.  In  this  position,  the  telescope  makes  with  the  normal  to 
the  glass  plate  an  angle  equal  to  that  of  complete  polarization. 
This  angle  is  then  measured  in  the  usual  way.  Its  tangent 
should  be  equal  to  the  index  of  refraction  of  the  glass  for  the  color 
used.  As  a  check  this  index  may  be  determined  in  the  usual  way. 
Care  must  be  taken  to  have  the  surface  of  the  glass  to  be  tested 
perfectly  clean. 

EXAMPLE 

The  solar  spectrum  was  allowed  to  fall  across  the  slit  end  of  the 
collimator  and  the  sodium  line  placed  on  the  slit.  The  slit  was 
then  opened  wide  and  the  light  allowed  to  fall  on  a  piece  of  the 
same  glass  whose  index  was  determined  with  the  interferometer 
(cf.  page  65).  The  polarizer  was  removed  from  the  collimator, 
and  the  analyzer  set  so  that  its  plane  of  transmission  was  hori- 
zontal. It  was  found  that  the  light  was  completely  cut  off  when 
the  angle  of  incidence  was  i  =  56°  32';  hence,  tan  i  =  1.513  =  /A. 

II.     VERIFY  FRESNEL'S  LAWS  OF  REFLECTION 

The  apparatus  and  adjustments  are  those  of  Experiment  I. 

MEASUREMENTS.— As  stated  above  this  is  easily  done  by  meas- 
uring the  rotation  of  the  plane  of  polarization  which  is  produced 
by  the  reflection.  Using  the  glass  plate  as  a  reflecting  surface,  as 
above,  allow  filtered  sunlight  to  fall  upon  it  through  the  collimator 
and  polarizer.  The  polarizer  should  then  be  set  at  a  definite 


154  MANUAL   OF   ADVANCED    OPTICS 

azimuth  by  the  graduated  circle  on  which  its  position  is  read, 
and  the  light  allowed  to  be  reflected  at  a  measured  angle  i.  The 
azimuth  of  the  reflected  light  is  then  read  on  the  circle  which 
gives  the  position  of  the  analyzer,  the  latter  being  set  so  as  to 
extinguish  the  light  reflected  from  the  prism.  Having  deter- 
mined the  index  of  refraction  in  the  previous  experiment,  and 
since  i  and  0  are  measured,  equation  (73)  furnishes  a  check  upon 
Fresnel's  reflection  equations.  In  case  it  is  found  difficult  to 
locate  exactly  the  angle  of  complete  polarization  these  observa- 
tions of  the  rotation  of  the  plane  of  polarization  may  be  used  to 
find  it,  for  it  is  that  angle  at  which  the  azimuth  of  the  reflected 
vibration  is  90°  from  the  plane  of  incidence.  Hence,  if  these 
observations  be  plotted  with  the  *'s  as  abscissae  and  the  0's  as 
ordinates,  the  resulting  curve  will  cross  the  axis  of  abscissae  at 
a  point  corresponding  to  the  angle  of  complete  polarization. 

EXAMPLE 

The  polarizer  was  set  so  that  its  plane  of  transmission  was 
at  an  azimuth  of  45°  with  the  vertical  plane.  The  light  used 
in  Experiment  I  was  then  allowed  to  fall  upon  the  reflecting  glass 
plate  at  measured  angles  of  incidence  i,  and  the  following  were 
observed  as  the  azimuths  of  the  reflected  light: 


30° 
35° 
40° 
45° 

50° 
55° 

60° 
65° 

70° 

75° 


e  (Oss.) 

0  (CALC.) 

33°  24' 

33°  36' 

28°  50' 

29°  3' 

23°  50' 

23°  36' 

17°  20' 

17°  14' 

10°  20' 

10°  9' 

2°  20' 

2°'  31' 

-5°  10' 

-5°  16' 

-13°  20' 

-12°  56' 

-20°  20' 

-20°  13' 

-27°  40' 

-27°  0' 

THE    REFLECTION    OF    POLARIZED    LIGHT  155 

The  observed  values  were  plotted  in  a  curve  with  the  z's  as 
abscissae.  The  curve  crosses  the  axis  at  the  point  56°  30',  which 
corresponds  to  the  angle  of  complete  polarization  (cf .  Experiment 
I).  The  calculated  values  were  obtained  with  the  help  of  equa- 
tion (73). 


XV 
METALLIC   KEFLECTION 

Theory 

Experiments  on  the  light  reflected  from  metallic  surfaces 
furnish  the  following  facts : 

1.  When  plane  polarized  light  is  reflected  from   a   metallic 
surface,  the  reflected  light  is  elliptically  polarized,  unless  the  plane 
of  polarization  of  the  incident  light  is  either  parallel  or  perpen- 
dicular to  the  plane  of  incidence. 

2.  Metallic  surfaces  do  not  possess  the  faculty  of  completely 
polarizing  light  by  a  single  reflection  at  any  angle. 

3.  If  the  incident   light  is  circularly  polarized,  there  is  one 
particular  angle  of  incidence  for  which  the  reflected  light  is  plane 
polarized. 

In  discussing  the  phenomena  of  metallic  reflection,  it  is  con- 
venient to  conceive  that  when  light  falls  upon  such  a  surface,  the 
incident  vibration  is  resolved  into  two,  one  perpendicular  and  one 
parallel  to  the  plane  of  incidence,  and  that  each  of  these  two 
components  undergoes  a  change  of  phase  at  reflection.  Experi- 
ment shows  that  that  component  which  is  parallel  to  the  plane  of 
incidence  undergoes  a  greater  change  of  phase  than  the  other,  and 
that  the  difference  in  phase  between  the  two  components  after 
reflection  is  zero  at  normal  incidence,  and  increases  with  the  angle 
of  incidence. 

Since  circularly  polarized  light  becomes  plane  polarized  by 
reflection  at  a  particular  angle  of  incidence,  this  angle  corresponds 
somewhat  to  the  angle  of  complete  polarization  of  transparent 

substances  and  is  called  the  principal  angle  of  incidence.     It  is 

156 


METALLIC    REFLECTION  157 

defined  as  the  angle  at  which  the  difference  in  phase  between  the 
component  in  the  plane  of  incidence  and  the  one  perpendicular  to 
it  amounts  to  a  quarter  of  a  wave  length.  This  particular  angle 
will  be  denoted  by  /.  The  azimuth  of  the  reflected  plane  polar- 
ized light  at  this  angle  of  incidence  is  called  the  principal  azimuth, 
and  will  be  denoted  by  0. 

The  theory  of  metallic  reflection  deduces  a  relation  between 
these  two  principal  angles  and  the  index  of  refraction  and  coeffi- 
cient of  absorption  of  the  metal.  The  index  of  refraction  /u,  of  the 
metal  is  defined  like  that  of  a  transparent  body  as  the  ratio  of  the 
velocity  of  light  in  vacuo  to  the  velocity  in  the  metal.  The 
coefficient  of  absorption  K  is  defined  as  follows :  If  A  represent 
the  amplitude  of  the  vibration  at  a  given  instant,  and  A'  its 
amplitude  after  the  wave  has  traveled  one  wave  length  in 
the  metal,  then  the  coefficient  is  defined  by  the  equation 
A  :  A'  =  1  :  e~2n*.  The  relation  between  these  optical  constants 
and  the  angles  /  and  6  is  expressed  by  the  equations* 

K  =  tan  20  1 

sin  /  tan  /  1(74) 

v/iT^ 

Hence  /w.  and  K  can  be  determined  from  observations  of  /  and  0. 

Experiment 

DETERMINE   THE   OPTICAL  CONSTANTS  OF  SILVER,  GOLD,   AND 

PLATINUM 

The  apparatus  and  adjustments  are  the  same  as  those 
described  in  Chapter  XIII. 

MEASUREMENTS. — The  measurements  may  be  made  in  two 
ways:  First,  we  may  allow  circularly  polarized  light  to  fall  on 
the  metallic  surface  and  determine  the  angle  of  incidence  at  which 

*Drude,  Tlieory  of  Optics,  p.  361  seq.,  Longmans,  1902.  Drude,  Wied. 
Ann.  36,  p.  885,  1889;  39,  p.  481,  1890. 


158  MANUAL    OF    ADVANCED    OPTICS 

the  reflected  light  is  plane  polarized,  and  the  azimuth  of  the  plane 
of  polarization. 

Second,  we  may  allow  light  plane  polarized  at  an  azimuth  of 
45°  to  fall  upon  the  metallic  surface,  and  observe  the  angle  of 
incidence  at  which  the  central  fringe  of  a  Babinet  compensator  set 
for  a  quarter-wave  returns  to  the  zero  position.  The  amount 
which  the  Nieol  behind  the  Babinet  has  to  be  rotated  to  make 
the  central  fringe  black  determines  the  principal  azimuth. 

The  metallic  surface  is  set  upon  the  prism  table  of  the  spec- 
trometer and  adjusted  to  be  perpendicular  to  the  telescope. 

EXAMPLE 

Glass  plates  coated  with  opaque  films  of  silver,  gold,  and 
platinum  were  used,  also  a  plate  of  polished  steel.  The  following 
observations  and  results  were  obtained,  the  values  of  /A  and  K 
being  computed  with  the  help  of  equations  (74) : 

METAL,  /  0  *  n 

Silver 74°  38'  43°  20'  17.2  0.20 

Gold 71°  40'  41°  50'  9.02  0.32 

Platinum 77°  30'  32°  40'  2.17  2,01 

Steel..... 76°     5'  28°  10'  1.50  2.56 

Sunlight  filtered  through  the  red  solution  described  in  Appendix  A 
was  used. 


XVI 

THE   SPECTROPHOTOMETER 
Theory 

Ordinary  photometers,  like  the  Bunsen  or  the  Lummer- 
Brodhun,  may  be  used  to  compare  the  intensities  of  the  total  radi- 
ations of  two  sources.  It  is,  however,  often  necessary  to  be  able 
to  compare  the  intensities  of  radiation  of  two  sources  for  each  of 
the  separate  colors.  This  is  accomplished  by  the  spectrophotom- 
eter  by  separating  each  of  the  sources  into  a  spectrum,  and 
arranging  the  optical  parts  so  that  the  two  spectra  are  adjacent 
and  can  be  compared  at  any  point  of  their  length. 

One  of  the  most  convenient  forms  of  spectrophotometer  is  that 
devised  by  Glan.*  This  instrument  consists  of  a  spectrometer 
whose  slit  is  divided  into  two  parts.  The  light  from  one  source  is 
allowed  to  pass  through  one  portion  of  the  slit,  while  that  from 
the  other  source  passes  through  the  other  portion.  We  thus 
get  in  the  field  of  view  two  adjacent  spectra,  one  from  each 
source.  A  screen  with  a  vertical  slit  in  it  in  the  eyepiece  cuts  off 
all  of  these  spectra  but  one  vertical  band  of  color,  half  of  which 
comes  from  one  source,  and  the  other  half  from  the  other. 

In  order  to  be  able  to  measure  the  relative  intensities  of  the 
two  halves  of  this  band  of  light,  the  apparatus  is  so  arranged 
that  the  light  in  one  half  is  polarized  in  one  plane  and  that  in  the 
other  in  a  perpendicular  plane.  This  is  accomplished  by  placing 
a  double-image  prism  in  the  collimator.  Such  a  prism  will  give 
two  images  of  each  half  of  the  slit,  and  these  two  will  be  polarized 

*Glan,  Wied.  Ann.  1,  p.  351,  1877. 

159 


160 


MANUAL    OF    ADVANCED    OPTICS 


in  planes  at  right  angles  to  each  other.  Thus  we  get  four  spectra 
in  the  field  of  view,  and  of  these  the  first  and  third  are  polarized  in 
one  plane,  while  the  second  and  fourth  are  polarized  in  a  perpen- 
dicular plane.  The  screen  in  the  eyepiece  cuts  off  the  first  and 
fourth  spectra  and  leaves  the  second  and  third,  one  from  the  upper 
half  of  the  slit  polarized,  say  in  a  vertical  plane,  and  one  from  the 
lower  half  of  the  slit  polarized  in  a  horizontal  plane.  By  intro- 
ducing a  Nicol  befcmd  the  double-image  prism  we  are  able  to  cut 
out  either  one  or  the  other  of  these  spectra.  Thus  when  the 
plane  of  transmission  of  the  Nicol  is  vertical,  the  spectrum  from 
the  upper  half  of  the  slit  will  be  cut  out;  while  when  that  plane 
is  horizontal,  the  spectrum  from  the  lower  half  of  the  slit  disap- 

w 


o 


FIGURE  40 


pears.  At  intermediate  positions  of  the  $Ticol,  both  spectra  are 
visible,  the  intensity  of  each  depending  on  the  intensities  of  the 
sources  and  the  position  of  the  Nicol. 

Let  /!  and  /2  represent  the  intensities  of  the  two  sources. 
Let,  further,  the  amplitude  of  vibration  of  the  light  from  source 
I  be  represented  by  OA  (Fig.  40),  and  that  of  source  2  by  OB. 
Suppose  also  that  OA  is  cut  out  when  the  index  on  the  Nicol 
stands  at  zero.  The  plane  of  transmission  of  the  Nicol  has  then 
the  direction  OB.  Conceive  the  Nicol  to  be  rotated  until  the  two 


THE    SPECTROPHOTOMETER  161 

spectra  are  equally  bright.  This  will  be  the  case  when  the  plane 
of  transmission  OC  has  such  a  direction  that  the  projections  of 
OA  and  OB  upon  it  are  equal,  i.e.,  when  it  is  perpendicular  to 
the  line  joining  A  and  B.  Call  the  angle  through  which  this 
plane  has  been  turned  a,  then  OB  cos  a  =  OC  =  OA  sin  a,  i.e., 
the  ratio  of  the  amplitudes  is 


\ 
OB     sin  a 


=  tan  a. 


0  A      cos  a 
Hence,  the  ratio  of  the  intensities  of  the  two  sources  is 

-  =  tan2  a.  (75) 


Experiments 

I.     COMPARE   THE   RADIATIONS   OF   Two  DIFFERENT   SOURCES 

OF  LIGHT 

APPARATUS.  —  The  spectrometer,  as  used  in  the  last  chapters, 
can  readily  be  converted  into  a  spectrophotometer. 

ADJUSTMENTS.  —  To  convert  the  spectrometer  into  a  spectro- 
photometer it  is  necessary,  as  stated  above,  to  place  in  the  colli- 
mator  tube  a  double-image  prism  directly  in  front  of  the  object- 
ive. This  prism  should  be  so  set  that  the  two  images  off  the 
vertical  slit  overlap  to  form  one  line.  The  vibrations  in  one  image 
will  then  be  vertical,  while  those  in  the  other  will  be  horizontal. 
A  narrow  card  should  then  be  fastened  over  the  middle  of  the 
slit  and  so  adjusted  in  width  that  the  overlapping  portions  of  the 
two  central  adjacent  images,  one  from  the  lower  and  one  from 
the  upper  half  of  the  slit,  are  cut  out.  The  center  of  the  image 
of  the  slit  in  the  eyepiece  then  appears  continuous,  but  the  vibra- 
tions of  the  upper  half  are,  say  vertical,  while  those  of  the 
lower  half  are  horizontal.  The  upper  and  lower  ends  of  the 
image  of  the  slit  should  then  be  cut  out  with  a  screen  in  the 


162  MANUAL    OF    ADVANCED    OPTICS 

eyepiece,  so  that  the  light  in  the  entire  upper  half  of  the  image 
vibrates  in  one  plane,  while  that  in  the  lower  vibrates  in  a  perpen- 
dicular plane. 

The  Nicol  is  then  put  in  place  in  the  tube  ~b  (Fig.  37)  at  the 
end  of  the  collimator.  The  index  is  set  at  zero,  and  the  Mcol 
rotated  in  the  collar  d  until  one  of  the  spectra  is  extinguished. 

A  prism  is  then  introduced  so  as  to  produce  in  the  field  of 
view  two  spectra,  adjacent,  'and  polarized  in  planes  at  right 
angles  to  each  other. 

MEASUREMENTS. — It  is  first  necessary  to  determine  a  scale  of 
wave  lengths.  To  do  this  allow  sunlight  to  pass  through  the  slit 
and  take  the  readings  on  the  circle  of  the  spectrometer  for  four  or 
five  of  the  Fraunhofer  lines.  Then  introduce  into  the  eyepiece  a 
screen  with  a  narrow  vertical  slit  whose  center  coincides  with  the 
intersection  of  the  cross-hairs.  Set  the  telescope  so  that  the  slit 
in  the  eyepiece  falls  upon  the  red  end  of  the  spectra  and  take  the 
reading  on  the  circle  of  the  spectrometer. 

Allow  the  two  lights  which  are  to  be  compared  to  enter 
the  two  halves  of  the  slit,  and  then  turn  the  Nicol  until  both 
halves  of  the  image  of  the  slit,  appear  equally  bright.  Take  the 
reading  of  this  position  of  the  Nicol.  Then  move  the  telescope 
along  a  definite  amount,  say  20'  or  30'.  Again  set  the  Nicol  for 
equal  illumination,  and  take  the  reading.  Proceed  in  this  way 
through  the  entire  spectrum.  The  readings  are  then  plotted 
with  the  angles  which  denote  the  successive  positions  of  the  tele- 
scope as  abscissae,  and  the  squares  of  the  tangents  of  the 
angles  of  the  Isicol  as  ordinates.  The  resulting  curve  repre- 
sents the  intensities  of  the  various  parts  of  the  spectrum  of 
one  source,  in  terms  of  those  of  the  other  as  unity.  For  if 
72  =  l,/i  =  tan2  a. 

Having  determined  the  angles  of  the  telescope  which  corre- 
spond to  certain  Fraunhofer  lines,  the  scale  of  abscissae  can  be 
converted  into  a  scale  of  wave  lengths. 


THE    SPECTROPHOTOMETER  163 

EXAMPLE 

Sunlight  was  used  to  calibrate  the  instrument  and  the  follow- 
ing readings  were  obtained: 

,FRAUNHOFER  LINE     READING 

B  14°    10' 

C  14°     0' 

D  13°  26' 

E  12°   32' 

F  11°  56' 

G  10°  36' 

An  incandescent  lamp  of  sixteen  candle  power  was  then  com- 
pared with  a  Welsbach  light.  The  following  readings  were 
obtained,  the  last  column  giving  tan3  a,  which  is  the  ratio  of  the 
intensity  of  the  Welsbach  to  that  of  the  incandescent  lamp : 

CORRESPONDING 


READING 

A  -  10—  « 

a 

TAN2  a 

11° 

0' 

444 

52° 

1.64 

20' 

458 

54° 

1.90 

40' 

473 

56° 

2.19 

12° 

0' 

490 

57° 

2.37 

20' 

508 

5T: 

2.37 

40' 

528 

56° 

2,19 

13° 

0' 

552 

55° 

2.04 

20' 

580 

54° 

1.90 

40' 

614 

50° 

1.42 

14° 

0' 

656 

46° 

1.08 

10' 

687 

38° 

0.61 

If  these  values  of  tan2  a  be  plotted  as  ordinates  with  either 
the  readings  or  the  corresponding  wave  lengths  as  abscissae,  the 
curve  will  show  that  the  Welsbach  is  relatively  richer  in  blue  and 
green  rays. 


164  MANUAL    OF    ADVANCED    OPTICS 

II.  DETERMINE   THE   ABSORPTION  OF  A  SOLUTION  OF   CYANIK 

Apparatus  and  adjustments  a's  in  Experiment  I. 

MEASUREMENTS. — Sunlight  is  allowed  to  pass  through  both 
halves  of  the  slit,  and  then  the  absorbing  substance  is  placed  over 
one  of  them.  The  method  of  making  and  plotting  the  observa- 
tions is  the  same  as  that  described  above.  The  student  should 
determine  several  such  absorption  curves  for  substances  like 
cyaiiin,  permanganate  of  potash,  ruby  glass,  or  a  thin  film  of 
silver  on  glass. 

EXAMPLE 

Light  from  an  incandescent  lamp  passed  into  the  upper  half 
of  the  slit  directly,  and  into  the  lower  half  through  a  glass  cell 
containing  a  dilute  solution  of  cyanin.  The  following  observa- 
tions were  made: 


READING 

a 

TAN2  a 

11° 

0' 

42° 

0.81 

12° 

0' 

42° 

0.81 

13° 

0' 

42° 

0.81 

10' 

31° 

0.36 

20' 

14° 

0.06 

30' 

0° 

0.00 

40' 

16° 

0.08 

50' 

38° 

0.61 

14° 

0' 

42° 

0.81 

If  these  values  of  tan2  a  are  plotted  as  ordinates  with  the  cor- 
responding readings  as  abscissae,  the  curve  will  represent  the 
intensity  of  the  light  transmitted  by  the  cyanin  solution  in  terms 
of  the  intensity  of  the  light  from  the  incandescent  lamp.  It  will 
be  noted  that  cyanin  absorbs  completely  the'radiation  correspond- 
ing to  the  reading  13°  30'.  From  the  preceding  example  it  is 
seen  that  this  reading  corresponds  to  wave  length  597  •  10"6  mm. 


XVII 
THE   DEVELOPMENT   OF   OPTICAL  THEORY 

By  optical  theory  is  meant  that  system  of  ideas  or  conceptions  f 
in  which  the  various  phenomena  of  light  are  unified,  and  by 
which  they  are  explained.  To  gain  a  clear  idea  of  the  meaning 
of  this  statement,  we  must  distinguish  two  factors  which  enter 
into  the  formation  of  every  science.  In  the  first  place,  we 
perceive  certain  external  events  through  their  effect  upon  our 
senses,  and,  in  the  second  place,  we  form  conceptions  by  which 
these  events  are  systematized,  harmonized,  and  interpreted. 

It  is  hardly  correct  to  apply  the  word  development  to  a  mere 
increase  in  the  number  of  phenomena  observed ;  for,  though  the 
number  of  things  which  we  perceive  with  regard  to  any  object 
becomes  greater  every  year,  this  increase  may  be  better  described 
by  the  word  accretion  than  by  the  word  development.  .  This  latter 
word  seems  to  imply  an  organized  increase,— an  evolution.  Hence 
it  can  appropriately  be  used  only  when  it  includes  a  reference  to 
our  conceptions,  for  it  is  in  the  organized  expansion  of  our 
knowledge  through  these  conceptions,  that  the  life  of  science 
really  lies. 

This  distinction  becomes  perfectly  clear  if  we  consider  an 
example.  Such  an  example  might  be  taken  at  random  from  almost 
any  domain  of  science;  but,  since  this  book  treats  of  optics,  it  will, 
perhaps,  be  better  to  draw  our  illustration  from  this  branch  of 
physics.  For  the  sake  of  brevity  we  will  begin  the  discussion 
with  the  end  of  the  seventeenth  century,  for  it  is  then  that  optics 
first  became  prominent. 

At  that  time  only  the  more  conspicuous  phenomena  of  light 
had  been  noted.  Thus,  it  was  known  that  light  seems  to  travel 

165 


166  MANUAL    OF    ADVANCED    OPTICS 

in  straight  lines,  that  it  is  reflected  from  a  plane  mirror  in  such  a 
way  that  the  angle  of  incidence  is  equal  to  the  angle  of  reflection, 
that  it  is  bent  from  its  straight  path  when  it  passes  obliquely 
from  one  medium  into  another  of  different  density,  and  that  the 
different  parts  of  a  beam  seem  to  be  independent  of  one  another. 

In  order  to  describe  these  phenomena  concisely,  two  concep- 
tions were  formed,  one  by  Descartes  and  Newton,  the  other  by 
Huygens  and  Hooke.  The  former  considered  that  light  consists 
of  fine  particles  or  corpuscles  which  are  shot  out  by  luminous 
bodies  and  travel  with  enormous  velocity ;  while  the  latter  believed 
that  light  is  a  form  of  wave  motion.  As  is  well  known,  the 
former  of  these  conceptions  prevailed  and  was  adopted  by  the 
scientists  of  the  eighteenth  century  as  being  more  nearly  correct. 
Newton  was  unable  to  adopt  the  latter  mainly  because  waves  are 
known  to  bend  or  be  diffracted  around  the  edges  of  obstacles 
placed  in  their  path,  and  the  light  waves  did  not  appear  to  do 
this.  Hence  he  lent  his  energies  to  developing  the  conception  of 
corpuscles  moving  in  straight  lines,  and  carried  with  him  the 
world  of  physicists  for  a  century  or  more. 

It  is  interesting  to  note  that  of  these  two  conceptions  that  of 
particles  moving  in  straight  lines  is  mechanically  much  simpler 
than  that  of  wave  motion,  for  it  is  characteristic  of  scientific 
thought  to  base  the  first  conception  of  a  phenomenon  upon  some- 
thing crudely  mechanical  or  which  is  familiar  and  easily  con- 
ceived. This  tendency  is  often  disastrous  to  the  healthy  progress 
of  a  subject,  and  proved  to  be  so  in  the  case  of  optics,  for  that 
science  made  very  little  advance  during  the  period  in  which  the 
conception  of  light  corpuscles  was  held. 

It  finally  became  evident  to  some  men  of  science  that  the  cor- 
puscular idea  must  be  abandoned,  first,  because  it  was  mechanic- 
ally too  simple  to  account  for  the  exceedingly  complex  and  varied 
phenomena  of  light;  and,  second,  because  it  seemed  to  them  to 
be  internally  absurd,  since  it  was  found  that  even  so  commonplace 


THE    DEVELOPMENT    OF    OPTICAL   THEORY  107 

an  event  as  the  passage  of  a  beam  of  light  from  air  into  water, 
for  example,  required  for  its  elucidation,  additional  assumptions 
which  were  somewhat  ridiculous,  as  shown  by  Xewton's  "fits  of 
easy  reflection, "  etc.  When  the  phenomena  of  interference  and 
polarization  became  matters  of  observation,  largely  through  the 
efforts  of  Young  and  Fresnel,  then,  notwithstanding  the  fact 
that  the  corpuscular  theory  had  to  adorn  itself  with  many  gratui- 
tous supplementary  hypotheses,  the  strife  between  the  rival  con- 
ceptions became  fierce,  and  all  parties  to  the  contest  sought  an 
''exper&nenttun  cruets"  which  would  finally  decide  between  them. 

Such  an  experiment  was  found  in  connection  with  refraction ; 
for,  according  to  the  corpuscular  theory,  light  must  travel  faster 
in  the  denser  medium ;  while,  according  to  the  wave  theory,  the 
reverse  is  true.  The  controversy  was  then  finally  settled  by 
Foucault  when  he  proved  experimentally  that  light  travels  more 
slowly  in  a  denser  medium,  and  so  the  conception  that  light  is  a 
form  of  wave  motion  came  to  be  generally  accepted. 

Thus,  although  the  corpuscular  theory  is  mechanically  the 
simpler,  the  phenomena  of  light  are  so  complex  that  the  more 
intricate  conception  of  wave  motion  has  after  all  proved  to  be 
more  satisfactory  in  that  it  not  only  furnished  simple  and  exact 
descriptions  of  interference,  diffraction,  and  polarization,  but  also 
enabled  the  scientists  of  that  day  to  predict  effects  which  led  to 
the  discovery  of  new  laws. 

It  was,  however,  soon  found  that  the  idea  that  light  is  a  wave 
motion  had  its  limitations  when  the  phenomena  of  dispersion 
were  considered.  In  dealing  with  diffraction  and  interference  it 
is  sufficient  to  conceive  that  we  have  to  do  with  a  wave  motion, 
but  in  the  case  of  dispersion  we  must  also  form  some  notion  of 
the  nature  of  the  medium  in  which  the  wave  motion  takes  place. 
Hence  the  next  problem  which  presented  itself  was  that  of  form- 
ing an  adequate  conception  of  that  medium, — the  ether,  as  it  has 
been  called. 


168  MANUAL    OF    ADVANCED    OPTICS 

The  most  natural  solution  of  this  problem  is  to  assume  that 
the  waves  are  elastic  waves  like  those  of  sound,  and  that  the 
medium  which  transmits  them  is  one  which  possesses  mechanical 
properties  similar  to  those  which  are  necessary  for  the  propagation 
of  elastic  waves.  If,  however,  this  solution  is  adopted,  several 
serious  difficulties  are  at  once  encountered. 

The  first  of  these  difficulties  arises  because  of  the  enormous 
velocity  of  light, — 300,000  kilometers  a  second.  Since  the 
velocity  of  an  elastic  impulse  in  any  medium  is  determined  by 
the  square  root  of  the  elasticity  of  the  medium  divided  by  its 
density,  it  is  necessary  to  assume  that  the  medium  which  trans- 
mits light  has  an  extraordinarily  high  elasticity,  or  a  very  small 
density,  or  both.  It  would  not  be  so  difficult  to  conceive  such  a 
medium  if  it  could  be  thought  of  as  a  very  rare  gas.  But  the 
light  waves  are  transverse  waves,  and  the  medium  which  transmits 
such  waves  must  have  rigidity,  i.e.,  must  have  the  properties  of  a 
solid. 

Even  this  conception  of  a  highly  elastic  and  very  rare  solid 
which  fills  all  space  might  not  be  impossible  if  it  were  not  for  the 
fact  that  the  planets  and  the  comets  move  at  great  velocities 
through  it  without  apparent  resistance,  and  the  conception  of  a 
solid  which  yet  offers  no  resistance  to  motion  through  it  is  rather 
difficult. 

But  there  are  other  conceptions  involved  in  the  assump- 
tion of  a  medium  which  reacts  to  mechanical  forces  which  are 
even  more  impossible.  These  .conceptions  depend  upon  the  elas- 
tic constants  of  the  medium.  The  theory  of  elasticity  demands 
that  six  conditions  be  fulfilled  when  an  elastic  impulse  passes  the 
boundary  between  two  media.  These  conditions  are  the  equality 
on  both  sides  of  the  boundary  of  the  components  of  the  elastic 
displacements,  and  the  equality  of  the  components  of  the  elastic 
forces.  In  order  to  satisfy  these  six  conditions  it  is  necessary  to 
assume  both  transverse  and  longitudinal  disturbances  in  the 


THE    DEVELOPMENT   OF    OPTICAL    THEORY  169 

second  medium,  for  the  transverse  alone  can  at  best  satisfy  four 
of  these  conditions.  Hence  it  has  been  the  burden  of  the  elastic 
theories  of  ether  to  make  these  four  constants  do  the  work  of  six, 
since  longitudinal  vibrations  in  the  ether  have  never  been 
detected.  A  detailed  account  of  the  devices  employed  by  Cauchy, 
Fresnel,  Green,  and  others  to  surmount  this  difficulty  will  be 
found  in  Winkelmann's  Handbuch  der  Physik,  Vol.  II,  pt.  1,  p. 
641  seq.* 

The  fact  which  is  of  special  interest  to  us  here,  is,  that 
although  the  adoption  of  the  idea  that  light  is  wave  motion  was 
a  step  toward  conceptions  of  greater  mechanical  complexity,  and 
hence  served  as  a  great  stimulus  to  progress  in  the  growth  of 
optical  theory,  yet  the  concomitant  notion  of  a  mechanically  elas- 
tic solid  medium  contained  internal  absurdities  which  placed 
serious  limitations  on  the  usefulness  of  the  theory  as  a  whole. 
Hence  it  appears  that  further  development  demanded  an  expan- 
sion or  change  in  the  conception  of  the  nature  of  the  ether. 

The  first  to  suggest  a  new  conception  was  Faraday,  \  who,  in 
1851,  while  discussing  the  question  whether  the  magnetic  force  is 
transferred  through  bodies  by  action  in  a  medium  external  to  the 
magnet  or  by  action  at  a  distance,  wrote:  "For  my  own  part,  con- 
sidering the  relation  of  a  vacuum  to  the  magnetic  force  and  the 
general  character  of  magnetic  phenomena  external  to  the  magnet, 
I  am  more  inclined  to  the  notion  that,  in  the  transmission  of  the 
force,  there  is  such  an  action  external  to  the  magnet,  than  that 
the  effects  are  merely  attraction  and  repulsion  at  a  distance. 
Such  an  action  maybe  a  function  of  the  ether;  for  it  is  not  at  all 
unlikely  that,  if  there  be  an  ether,  it  should  have  other  uses  than 
simply  the  conveyance  of  radiations." 

*  Cf.  also  Lloyd,  Report  on  Optical  Theories,  B.  A.  Reports,  1834,  p. 
295  seq.  Glazebrook,  Report  on  Optical  Tlieories,  B.  A.  Reports,  1885,  p. 
157  seq.  L'Abbe  Moigno,  Repertoire  d'Optique  Moderne,  4  Vols.,  Paris, 
1847-50. 

f  Faraday,  Experimental  Researches,  No.  3075. 


170  MANUAL    OP    ADVANCED    OPTICS 

This  hint  of  Faraday's  was  taken  up  later  by  Maxwell,*  who  in 
1873  published  a  theory  of  light  based  on  the  assumption  that 
the  medium  which  transmits  light  is  the  same  as  that  which 
serves  as  a  vehicle  for  the  electric  and  magnetic  forces.  He 
further  developed  a  conception  of  the  process  of  ether  wave 
propagation  .which  renders  this  theory  free  from  the  internal 
absurdities  and  contradictions  which  hindered  the  progress  of  its 
predecessor. 

Thus  his  fundamental  assumption  concerning  the  nature  of 
the  forces  in  the  ether,  i.e.,  his  conception  of  displacement  cur- 
rents which  produce  magnetic  effects,  is  such  that  it  follows  at 
once  from  it  that  electromagnetic  waves  in  the  ether  are  trans- 
verse, f  Hence  after  adopting  this  theory  we  are  no  longer  com- 
pelled to  consider  the  ether  a  solid. 

Furthermore,  there  are  only  four  independent  conditions 
which  must  be  fulfilled  when  a  train  of  waves  passes  through  the 
boundary  between  two  media,  and  hence  it  is  not  necessary  to 
make  any  special  assumptions  to  explain  that  passage. 

Finally,  the  theory  enables  us  to  calculate  optical  constants 
from  electrical  measurements,  thus  bringing  two  distinct  fields  of 
investigation  into  relations  which  can  be  subjected  to  quantitative 
measurements. 

Now,  although  the  attempt  to  describe  the  properties  of  this 
two-sided  medium,  the  ether,  as  if  they  were  similar  to  the  crudely 
mechanical  properties  of  grosser  matter,  has  since  then  fre- 
quently been  made,  science  is  fast  coming  to  believe,  if  it  has  not 
already  reached  the  conclusion,  that  this  ether  belongs  in  a  cate- 
gory by  itself, — that  its  properties  are  discretely  different  from 
those  of  perceptible  matter.  Thus  Maxwell's  theory  has  proved 
to  be  a  step  from  the  mechanical  conception  of  an  elastic  solid  to 
a  wholly  new  and  radically  different  idea.  It  thus  opened  at  once 

*  Ma,xwell^Electrieity_Qnd  Magnetism,  Vol.  II.,  p.  431. 
fDruule,  Theory  oj  Optics.,  p.  278. 


THE    DEVELOPMENT    OF    OPTICAL   THEORY  171 

a  vast  domain  for  new  investigation — it  bridged  the  chasm 
between  the  territories  of  light  and  electricity,  and  has  been  the 
means  of  adding  immensely  to  our  store  of  information  concern- 
ing both  sciences. 

It  is  to  be  noted,  however,  that  this  conclusion  that  the  ether 
is  discretely  different  from  ordinary  matter  does  not  by  any  means 
indicate  that  the  problem  is  completely  solved.  It  merely  places 
that  medium  in  a  position  in  which  an  open  discussion  of  its 
properties  is  possible,  i.e.,  it  removes  all  crudely  mechanical  bias 
from  our  minds  and  shows  us  that  we  have  before  us  an  almost 
wholly  unexplored  domain  which  invites  investigation.  It  thus 
lends  enthusiasm  to  optical  and  electrical  research  by  offering  a 
field  for  the  exercise  of  untrammeled  imagination, — a  field  unob- 
structed by  conceptions  of  a  grossly  mechanical  nature.  Some  of 
the  results  of  this  impetus  which  the  electromagnetic  theory  of 
light  has  given  to  optical  research  will  be  briefly  discussed  in  the 
next  chapter. 

In  brief,  then,  optical  theory  has  developed  from  a  simple  con- 
ception of  material  particles  traveling  in  straight  lines,  through 
the  more  complex  conception  of  waves  in  a  mechanically  elastic 
medium,  to  the  recognition  of  a  wave  motion  in  a  medium  whose 
properties  are  still  to  be  discovered  and  classified.  And  while 
the  theory  has  passed  through  these  stages,  the  observed  details 
in  the  phenomena  which  it  is  invented  to  describe  have  increased 
in  number  in  a  way  commensurate  with  the  expansion  of  the 
theory.  And  so  we  return  to  the  statement  with  which  we 
began,  namely,  that  science  is  a  vitally  living  thing,  whose  life  can 
be  traced  in  the  development  of  the  conceptions  which  we  form 
of  phenomena  in  the  world  about  us. 


XVIII 
THE   TREND    OF   MODERN    OPTICS 

It  remains  for  us  to  consider  briefly  some  of  the  more  marked 
results  to  which  the  adoption  of  the  electromagnetic  theory  has 
led.  The  first  important  step  in  the  development  of  the  theory 
was  taken  by  Maxwell  when  he  showed  that  the  velocity  of  an 
electromagnetic  wave  in  a  dielectric  is  equal  to  the  ratio  of  the 
electromagnetic  to  the  electrostatic  unit,  jdivijiejiJay^b^^sqiiaxe 
root  of  the  dielectric  constant  of  frhe  medium.  Thus  if  V  repre- 
sent the  velocity  of  the  wave,  c  the  ratio  of  the  units,  and  k  the 
dielectric  constant,* 

v-j%  (™) 

Two  important  results  follow  from  this  equation.  The  first  is 
derived  from  the  fact  that  the  dielectric  constant  of  the  ether  is 
defined  as  unity.  Hence  the  velocity  of  an  electromagnetic  wave 
in  the  ether  is  equal  to  the  ratio  of  the  electromagnetic  to  the 
electrostatic  units.  Now  this  ratio  of  the  units  has  been  deter- 

pm 
mined  by  experiment  and  found  to  have  the  value  3  •  1010  — -• 

sec. 

But  this  is  the  velocity  of  light  in  the  free  ether,  in  fact  the  two 
numbers  agree  so  closely  that  we  can  hardly  regard  it  as  a  mere 
coincidence,  but  are  led  to  believe  that  light  is  an  electromagnetic 
vibration. 

The  second  result  relates  to  the  index  of  refraction.  In  Chap- 
ter VII  this  index  has  been  defined  as  the  ratio  of  the  velocity  of 
light  in  ether  to  its  velocity  in  the  medium  considered.  Thus  if 

*Drude,  Theory  of  Optics,  p.  276,  Longmans,  1902. 

172 


THE   TREND    OF   MODERN    OPTICS  173 

V  represent  the  velocity  in  the  medium,  k'  the  dielectric  con- 
stant, and  fji  the  index  for  infinitely  long  waves,  we  have  from 
equation  (76),  since  k  =  1, 


i.e.,  the  square  of  the  index  is  equal  to  the  dielectric  constant. 
This  conclusion  has  also  been  tested  by  experiment.  The  first 
results  were,  however,  rather  discouraging,  for  it  was  found  that, 
while  the  relation  proved  true  for  some  substances,  it  was  far 
from  correct  for  others.  Thus  for  benzole  we  find  /x,  =  1.482,  and 
y/k'  =  1.49;  while  for  water,  /u.  =  1.33  and  <STc'  =  9.0. 

It  was  some  years  before  this  discrepancy  was  explained  in  a 
satisfactory  way.  It  was  suggested  that  the  difficulty  might  be 
due  to  the  fact  that  vibrations  of  short  period  were  used  in  deter- 
mining the  index  for  the  visible  spectrum  from  which  the  index 
for  infinitely  long  waves  was  calculated  from  a  dispersion  equa- 
tion like  that  of  Cauchy  (cf.  p.  91),  while  very  long  oscillations 
were  used  in  measuring  k' .  The  objection  to  this  explanation 
was,  that  when  there  is  a  discrepancy,  k'  is  always  greater  than 
/x2,  whereas,  according  to  the  experience  of  that  time,  the  longer 
the  wave,  the  smaller  the  index ;  so  that  for  infinitely  long  waves 
the  index  should  be  even  smaller  than  it  is  for  the  visible  spec- 
trum, thus  making  the  discrepancy  worse. 

It  was  at  length  discovered  that  the  dispersion  does  not  always 
follow  an  equation  like  that  of  Cauchy;  in  short,  it  was  found 
that  when  the  medium  absorbs  some  of  the  radiations  which  fall 
upon  it,  the  dispersion  law  is  quite  different.  Thus  if,  for  exam- 
ple, a  spectrum  be  formed  by  a  prism  of  cyanin,  which  absorbs  the 
yellow,  it  will  be  found  that  the  red  and  orange  rays  are  deflected 
more  than  the  green,  blue,  and  violet,  the  two  halves  being  separa- 
ted by  a  dark  band  where  the  yellow  should  be.  Such  dispersion 
is  called  anomalous.  From  this  discovery  it  became  at  once  clear 


174  MANUAL    OP    ADVANCED    OPTICS 

that  we  can  not,  in  the  case  of  absorbing  media,  calculate,  with 
the  help  of  the  dispersion  equations  which  belong  to  transparent 
media,  the  numerical  value  of  the  index  of  refraction  on  one  side 
of  an  absorption  band  from  observations  taken  on  the  other  side. 

After  this  it  was  easy  to  surmise  that  substances  like  water, 
for  which  Tcf  is  greater  than  /x2,  have  absorption  bands  which 
correspond  to  periods  of  vibration  between  those  of  visible  light 
and  those  by  which  k'  is  measured,  i.e.,  beyond  the  ultra  red. 
To  prove  this  it  is  merely  necessary  to  measure  the  index  of 
refraction  for  very  long  oscillations.  This  was  done  for  water 
by  Drude,*  and  the  result  was  a  complete  confirmation  of 
the  theory,  for  he  found  that  for  such  long  waves  the  index  of 
refraction  of  water  has  the  value  9.0.  Hence  the  original  state- 
ment of  Maxwell,  namely  /*  =  \/&',  has  been  modified  to  read  /x 
can  not  be  greater  than  */k'.  If  for  any  substance  it  is  less,  we 
believe  that  we  can  with  certainty  predict  that  that  substance 
absorbs  completely  some  of  the  radiations  which  correspond  to 
the  ultra  red. 

But  this  discovery  of  the  relation  between  absorption  and 
anomalous  dispersion  had  an  even  greater  result  than  that  of 
clearing  away  the  discrepancy  in  Maxwell's  original  theory,  for 
the  question,  why  should  a  substance  absorb  one  particular  set 
of  waves,  led  for  its  answer  to  the  conception  that  the  smallest 
particles  of  such  absorbing  media  must  be  free  to  vibrate,  and 
have  natural  periods  of  vibration  which  are  the  same  as  those  of 
the  waves  which  they  absorb.  This  conception  was  borrowed 
from  the  known  phenomena  of  absorption  by  vapors  as  mani- 
fested in  the  Fraunhofer  line§  in  the  solar  spectrum.  But  the 
assumption  that  the  smallest  particles  are  set  in  vibration  by  an 
electromagnetic  wave  whose  period  coincides  with  their  own  nat- 
ural period  involves  the  idea  that  such  particles  must  carry  elec- 

*  Drude,  Wied.  Ann.  59,  p.  17,  1896. 


THE    TREND    OF   MODERN    OPTICS  175 

trie  charges.  Hence  we  reach  a  conception  which  is  one  of  the 
most  important  in  modern  science,  namely,  this:  That  the 
particles  of  all  substances  which  allow  light  to  pass  at  all  are 
charged  particles  which  have  definite  natural  periods  of  vibration. 
Such  charged  particles  have  been  given  the  name  ions. 

This  conception  is  not  peculiar  to  optics,  but  is  found  to  be 
necessary  in  describing  the  phenomena  of  electrolysis,  and  has 
proved  itself  equally  indispensable  in  discussing  the  cathode  rays 
and  the  action  of  magnetism  on  light. 

But  how  does  this  idea  help  our  understanding  of  the  facts  of 
normal  dispersion?  Because  it  can  be  shown*  that  the  dielectric 
constant  of  a  medium  composed  of  charged  particles  which  have 
natural  periods  of  vibration,  depends  upon  the  ratio  of  those  nat- 
ural periods  to  that  of  the  impressed  vibration.  When  the  value 
of  this  ratio  is  not  unity,  i.e.,  when  the  impressed  period  and  the 
natural  periods  are  not  the  same,  the  dielectric  constant  varies 
continuously  with  the  period,  so  that  the  curve  which  expresses 
the  relation  between  the  two  is  continuous.  Hence  the  curve 
which  expresses  the  relation  between  the  index  of  refraction  and 
the  wave  length  is  also  continuous,  i.e.,  it  agrees  with  the  observed 
dispersion  curve.  When,  however,  the  impressed  and  the  natural 
periods  coincide,  then  absorption  takes  place,  and  the  dispersion 
seems  to  become  discontinuous  at  that  point. 

But  this  idea  of  charged  particles  which  have  natural  periods 
can  be  extended  so  as  to  give  us  some  conception  of  the  nature  of 
the  particle.  For  it  is  not  difficult  to  conceive  that  if  by  any 
means  we  could  set  those  ions  into  violent  vibrations,  they  would 
become  a  source  of  electromagnetic  waves  of  the  same  period  as 
the  natural  periods  of  the  particles.  Now  it  is  a  well-known  fact 
that  all  solids,  when  heated  to  incandescence,  send  out  waves  of 
all  possible  periods ;  but  that  when  the  solid  has  been  changed  to 

*  Drude,  Theory  of  Optics,  p.  382. 


176  MANUAL    OF    ADVANCED    OPTICS 

a  vapor,  the  vibrations  sent  out  have  certain  particular  periods 
which  are  peculiar  to  the  substance  and  characteristic  of  it. 
These  facts  are  the  basis  of  spectrum  analysis,  and  lead  to  two 
conceptions  which  bear  closely  upon  the  general  theory  of  optics. 

For  when  these  facts  are  considered  in  relation  to  the  differ- 
ence in  the  volumes  occupied  by  the  same  quantity  of  a  substance 
in  the  solid  and  in  the  vapor  state,  we  must  recognize  that  the  ions 
are  much  closer  together  when  they  send  out  white  light  than 
they  are  when  they  produce  only  certain  definite  vibrations. 
Hence  the  conception  has  been  formed  that  the  ions,  which  by 
their  vibration  produce  light,  are  grouped  into  larger  particles. 
When  the  substance  is  in  a  solid  state  and  heated  to  incandes- 
cence, these  larger  particles  are  set  into  comparatively  slow  vibra- 
tion. But  since  they  are  close- t-ogether,  they  will  collide  with  one 
another  very  frequently,  i.e.,  each  particle  will  not  travel  very  far 
before  it  meets  another.  At  every  such  collision  between  two 
particles  each  receives  a  blow  which  causes  it  to  shiver  so  that  the 
ions  which  compose  it  are  set  into  violent  vibration.  Now  these 
vibrations  due  to  impact  are  not  at  first  those  corresponding  to 
the  natural  period  of  the  ion,  but  are  forced  vibrations.  Hence 
they  may  have  any  period  whatever.  Such  forced  vibrations, 
however,  die  down  very  rapidly,  and  the  ions  would  then  continue 
to  vibrate  in  their  natural  periods  were  it  not  for  the  fact  that 
they  are  so  close  together  that  the  forced  vibrations  do  not  have 
time  to  disappear  between  impacts.  Hence  white  light  may  be 
conceived  to  be  due  to  the  vibrations  forced  upon  the  ions  by 
the  frequent  impact  of  the  particles  in  which  they  are  grouped. 

But  when  the  substance  is  in  the  form  of  vapor,  the  particles 
are  farther  apart,  so  that  each  particle  travels  very  much  farther 
between  impacts.  In  this  case  the  time  required  for  the  forced 
vibrations  to  die  away  is  small  in  comparison  with  the  time  between 
impacts,  so  that  the  ions  send  out  waves  of  their  natural  periods 
mainly.  Hence  the  spectrum  is  no  longer  continuous  but  is  com- 


THE   TREND    OF   MODERN    OPTICS  177 

posed  of  a  few  lines  only,  namely,  of  those  which  correspond  to 
the  natural  periods  of  the  ions. 

Now  we  know  that  a  simple  sounding  body,  like  a  vibrating 
string,  sends  out  a  series  of  vibrations,  partial  vibrations  as  they 
are  called,  whose  periods  stand  related  to  one  another  inversely  as 
the  numbers  1,  2,  3,  etc.  When  the  geometrical  form  of  the 
sounding  body  is  not  so  simple,  the  relation  between  the  partial 
vibrations  becomes  more  complex;  in  fact,  the  theoretical  solu- 
tion of  the  case  of  even  so  simple  a  form  as  the  ring  has  not  yet 
been  worked  out.  Now  in  simple  cases  we  can  analyze  the  com- 
pound sound  vibrations  which  come  to  us  into  the  partial  vibra- 
tions, and  from  that  analysis  draw  certain  conclusions  as  to  the 
nature  of  the  sounding  body.  Can  we  do  the  same  with  the  light 
vibrations?  Can  we  find  a  numerical  relation  between  the  periods 
of  vibration  indicated  by  the  bright  lines  in  the  spectrum  of  a 
substance,  and  from  that  relation  draw  conclusions  as  to  the 
geometrical  form  of  the  vibrating  ions?  This  is  an  extremely 
interesting  and  pertinent  question,  for  the  determination  of  the 
shape  of  what  has  been  called  an  atom  has  been  for  ages  one  of 
the  favorite  problems  of  the  philosophers. 

Now  experiment  has  shown  that  there  is  a  numerical  relation 
between  the  periods  of  vibration  of  the  lines  of  each  of  a  large 
number  of  spectra.  That  this  is  not  mere  chance  is  shown  by  the 
fact  that  this  numerical  relationship  has  led  to  the  discovery  of 
new  lines  whose  position  in  the  series  had  been  calculated  but 
which  had  not  before  been  observed.  Investigation  has  also 
established  relationships  between  the  groups  of  the  Mendeleeff 
series.  Thus  the  elements  of  the  first  and  third  groups  (Li. ,  Na. , 
K.,  Rb.,  Cs.,  Mg.,  Ca.,  Sr.)  are  characterized  by  doublets,  while  the 
elements  of  the  second  group  (Cu.,  Ag.)  show  series  of  triplets. 
But,  although  an  equation  has  been  found*  which  expresses 

*Kayser  and  Runge,  Abh.  d.  k.  Preuss.,  Akad.  d.  Wiss.,  1888-1893. 
Winkelruann's  Handbuch,  II,  1,  p.  441  seq. 


178  MANUAL    OF    ADVANCED    OPTICS 

numerically  the  relation  between  the  periods  of  vibration,  the 
geometrical  form  of  the  object  that  would  be  capable  of  sending 
out  such  a  series  of  partial  vibrations  has  not  yet  been  deter- 
mined. 

The  problem  is,  however,  by  no  means  hopeless,  and  every  new 
detail  discovered  in  the  spectrum  of  any  substance  adds  its  share, 
be  that  little  or  great,  to  the  general  store  of  information.  The 
importance  of  the  detailed  study  of  spectra  is  thus  made  clear, 
for  the  more  detailed  our  knowledge  of  the  vibrations  of  the  ions, 
the  nearer  we  come  to  being  able  to  form  a  conception  of  their 
size  and  structure.  Hence  the  value  of  increasing  in  every  way 
the  power  of  spectroscopic  apparatus  becomes  manifest,  and  we 
can  appreciate  the  bearing  of  the  chapter  on  visibility  curves,  for 
this  is  at  present  the  most  refined  method  of  light  wave  analysis, 
i.e.,  the  method  which  furnishes  the  most  detailed  information  as 
to  the  exact  nature  of  any  complex  set  of  light  vibrations. 

Another  factor  which  helps  to  emphasize  the  usefulness  of 
increased  spectroscopic  power  is  the  recent  discovery  of  the  action 
of  magnetism  on  the  vibrations  of  the  ions.  Faraday  had  sus- 
pected that  the  magnetic  field  should  act  upon  the  vibrations  of 
light,  and  discovered  the  magnetic  rotation  of  the  plane  of  polari- 
zation, but  he  was  unable  to  find  that  the  field  produced  any 
change  in  the  period  of  vibration  of  the  light.  The  effect  was 
discovered  a  few  years  ago  (1897)  by  Zeeman,  who  found  that 
each  line  of  a  spectrum  seems,  when  the  source  of  light  is  placed 
between  the  poles  of  a  powerful  magnet,  to  be  separated  into 
three  lines,  the  two  outer  being  polarized  in  one  plane,  and  the 
middle  one  in  a  plane  at  right  angles. 

The  theoretical  explanation  of  this  phenomenon  is  as  simple 
as  it  is  beautiful.  For  we  have  come  to  believe,  as  has  been 
stated,  that  the  light  waves  are  caused  by  vibrating  ions,  and,  in 
the  general  case,  the  paths  which  such  vibrating  ions  follow  are 
ellipses.  But  an  elliptical  motion  can  be  resolved  into  a  circular 


THE   TREND   OF   MODERN   OPTICS  179 

motion  in  one  plane  and  a  rectilinear  motion  in  a  plane  at  right 
angles.  Hence  we  may  assume  that  the  vibration  forms  of  the 
ions  are  thus  resolved  into  circles  in  planes  at  right  angles  to  the 
lines  of  force  of  the  field,  and  straight  lines  parallel  to  those 
lines.  Xow  a  moving  electric  charge  is  equivalent  to  an  electric 
current,  and,  as  is  well  known,  a  magnet  attracts  or  repels  an 
electric  current  which  flows  at  right  angles  to  the  lines  of  force  of 
the  field  according  to  the  direction  of  the  current.  Since  we 
have  resolved  the  vibrations  into  circles  whose  planes  are  perpen- 
dicular to  the  lines  of  the  field  and  straight  lines  parallel  to  those 
lines,  only  the  circular  vibrations  will  be  affected  by  the  field. 
But  since  some  of  the  ions  are  rotating  in  one  direction  and  others 
in  the  opposite  direction,  the  effect  of  the  field  upon  them  will  be 
different  in  the  two  cases.  For  if  the  field  act  upon  those  rotating 
in  one  direction  to  repel  the  ion,  it  will  act  with  an  attraction  upon 
those  rotating  in  the  opposite  direction.  If,  therefore,  in  the  first 
case,  the  circular  path  of  the  ion  is  made  larger,  and  hence  the 
period  of  vibration  increased,  in  the  second  case  the  path  will  be 
contracted,  reducing  the  period  of  the  oscillation.  Since  these 
vibrations  are  in  the  plane  perpendicular  to  the  lines  of  force  of 
the  field,  they  will  appear  to  an  observer  looking  at  right  angles 
to  those  lines  like  plane  vibrations.  Hence,  if  the  lines  of  force 
be  horizontal,  these  vibrations  will  appear  to  be  vertical,  and  the 
observer  will  perceive  two  plane  polarized  vibrations  whose  periods 
are  slightly  different  from  that  of  the  original  vibration,  one  being 
somewhat  smaller,  the  other  somewhat  greater. 

But  the  period  of  the  vibrations  parallel  to  the  lines  of  force 
of  the  field  is  not  altered  by  the  field.  Hence  the  observer  should 
also  perceive  a  plane  polarized  vibration  whose  oscillations  are 
horizontal.  Hence  we  can  conceive  how  a  single  line  can  be  sep- 
arated into  three,  in  two  of  which  the  vibrations  are  vertical  and 
the  period  of  vibration  altered,  while  in  the  third  the  vibrations 
are  horizontal  and  of  the  same  period  as  that  of  the  original  light. 


180  MANUAL   OF    ADVANCED    OPTICS 

If,  however,  an  observer  looks  along  the  lines  of  force  of  the 
field  he  should  perceive  two  circular  vibrations  rotating  in  oppo- 
site directions.  Observation  confirms  this  conclusion  also. 

But  while  observation  confirms  in  general  these  results,  it  has 
nevertheless  been  found  that  when  the  analysis  of  the  vibrations 
is  carried  farther,  the  phenomenon  is  not  so  simple.  For  it  has 
been  shown*  that  a  single  line  is  not  merely  separated  into  three, 
as  has  been  described,  but  that  each  of  the  three  may  itself  be 
compound.  Furthermore,  the  complexity  of  the  lines  is  different 
for  different  lines  in  the  spectrum  of  the  same  element.  Thus  in 
the  case  of  cadmium  the  three  lines  into  which  the  red  radiation 
is  separated  are  all  single ;  while  the  green  radiation  breaks  into  a 
triplet  whose  vibrations  are  horizontal  and  two  quadruplets  whose 
vibrations  are  vertical ;  and  in  the  case  of  the  blue  radiation,  all 
the  lines  are  doublets.  These  details  have  not  yet  been  satisfac- 
torily accounted  for  in  the  theory.  The  entire  subject  is  one  of 
great  interest,  and  promises  well  to  become  an  important  factor 
in  helping  us  to  form  conceptions  of  the  geometrical  construction 
of  the  ions.  Since  the  effects  here  described  are  so  small  that 
they  can  be  detected  only  by  the  highest  spectroscopic  power  at 
present  available,  the  value  of  increasing  the  resolving  power  of 
the  spectroscope  becomes  again  manifest.  A  very  decided 
advance  has  recently  been  made  in  this  direction  by  Michelsonf  in 
the  invention  of  his  echelon  spectroscope. 

Another  very  interesting  conclusion  has  been  reached  from  the 
consideration  of  this  effect  of  the  magnetic  field  upon  the  vibra- 
tion of  ions.  For  it  is  clear  that  the  magnitude  of  these  effects 
depends  upon  two  factors,  namely,  upon  the  strength  of  the 
electric  charge  carried  by  the  ion,  and  upon  the  mass  of  the 
ion.  Thus  the  greater  the  charge,  the  greater  will  be  the 

*Michelson,  Phil.  Mag.  (5)  44,  P-  109,  1897;  45,  p.  348,  1898. 
fMichelson,   Am.   Jour.  Sci.  (4)  5,  p.  215;  Astrophys.  Jour.  8,  p.  37, 
1898. 


THE   TREND    OF   MODERN    OPTICS  181 

effect,  and  the  greater  the  mass,  the  smaller  will  be  the  effect. 
Hence  if  e  represent  the  charge,  and  m  the  mass  of  the  ion,  the 

/> 

effect  will  be  proportional  to  — —    Now  there  are  other  phenomena 

in  the  description  of  which  this  factor  -  -  appears,  namely,  elec- 
trolysis and  the  cathode  rays.  For  in  the  case  of  the  former  the 
action  will  evidently  be  more  rapid  the  greater  the  charge  and 
the  less  the  mass  of  the  ion;  and  the  deflection  of  the  cathode 
rays  by  the  magnetic  field  will,  upon  the  assumption  that  those 
rays  are  caused  by  ions  shot  out  from  the  cathode,  be  greater  the 
greater  the  charge  and  the  less  the  mass  of  the  ion. 

Unfortunately  these  phenomena  do  not  permit  of  an  inde- 
pendent determination  of  either  e  or  m,  but  lead  only  to  a  value 
of  their  ratio.  Xow  the  value  of  this  ratio  for  hydrogen,  as 
determined  from  electrolysis,  is  about  10*,  while  its  value,  as 
determined  from  the  effects  of  the  magnetic  field  upon  the 
cathode  rays  and  the  light  vibrations,  is  about  1.7  •  107.  The 
difference  in  the  two  cases  may  be  accounted  for  in  three  ways : 
namely,  either  e  is  the  same  in  the  three  cases,  while  m  is  greater 
in  electrolysis;  or  the  /?i's  are  the  same,  but  the  e's  are  different 
in  the  three  cases;  or  both  e  and  m  vary  in  the  different  phe- 
nomena. Hence  it  is  an  interesting  problem  to  find  a  means  of 
obtaining  independent  determinations  of  the  two  quantities. 

This  has  not  yet  been  done  without  making  assumptions  as  to 
the  number  of  particles  in  a  given  volume  at  a  given  pressure. 
But  it  is  found  that,  if  we  take  for  that  number  the  one  to  which 
the  kinetic  theory  of  gases  leads  us,  we  find  as  the  charge  of  the 
univalent  ion  in  electrolysis  1.29  •  10~10.*  J.  J.  Thomson  f  has 
calculated  from  observations  upon  ions  in  vacuo  a  number  of  the 

*  Drude,  Theory  of  Optics,  p.  532. 

t  J.  J.  Thomson,  Phil.  Mag.  (5)  44,  p.  293,  1897;  46,  p.  528,  1898;  £?, 
p.  547,  1899. 


182  MANUAL   OF   ADVANCED   OPTICS 

same  order,  namely,  6.7  •  10~10,  as  the  value  of  the  charge  upon  an 
ion  under  those  conditions.  Hence  the  conclusion  seems  probable 
that  the  charges  are,  in  the  two  cases,  nearly  the  same,  but  that 
the  masses  of  th$  ions  in  electrolysis  are  greater  than  those  of  the 
ions  which  take  part  in  the  phenomena  of  the  cathode  rays  and  of 
the  action  of  magnetism  on  light. 

Hence  physicists  have  come  to  believe  that  the  particles  whose 
vibrations  cause  light  are  parts  of  an  atom, — that  the  time-hon- 
ored atom,  which  to  the  chemist  is  indivisible,  is  in  reality  com- 
pounded of  a  veiy  large  number  of  smaller  ions  to  which  the 
special  name  electron  has  been  given.  The  calculations  are  not 
exact,  but  there  is  good  reason  for  the  belief  that  the  hydrogen 
atom  is  made  up  of  something  like  one  thousand  of  these  elec- 
trons, and  that  the  numbers  in  the  other  chemical  atoms  are 
roughly  proportional  to  the  atomic  weights.  According  to  this 
conception  an  atom  of  mercury  would  be  composed  of  about 
200,000  electrons. 

Such  ideas  as  these  tax  the  powers  of  conception  of  the  human 
mind  to  the  limit.  For,  if  it  is  difficult  to  form  a  mental  image 
of  an  atom,  which  is  so  minute  that  we  can  not  even  with  the  best 
microscope  see  a  group  of  several  hundred  of  them,  is  it  not  bold 
of  us  to  attempt  to  realize  that  such  atoms  are  themselves  highly 
complex? 

Hence  we  can  see  that  the  science  of  optics  is  not  a  worn-out, 
barren  field  of  investigation,  but  that  it  now  throbs  with  vigorous 
life,  and  that  the  conceptions  of  Nature  which  it  has  developed 
under  the  stimulus  of  the  electromagnetic  theory  have  led,  and 
probably  will  always  lead,  the  human  mind  to  the  very  outermost 
boundaries  of  the  knowable  and  the  conceivable. 


APPENDIX 

A.    SOURCES  OF  LIGHT 

One  of  the  most  useful  sources  of  monochromatic  light  is  the 
sodium  flame.  Such  a  flame  can  readily  be  produced  by  heating  a 
piece  of  hard  glass  tubing  in  the  flame  of  a  Bunsen  burner.  The 
tubing  can  be  supported  in  the  flame  by  a  piece  of  iron  wire,  the 
other  end  of  the  wire  being  wound  around  the  burner.  Or  a  piece 
of  asbestos  which  has  been  soaked  in  a  strong  solution  of  one  of 
the  sodium  salts  can  be  tied  about  the  upper  end  of  the  burner. 
This  sort  of  flame  is  very  convenient  in  the  interferometer  work, 
especially  if  there  be  mounted  on  the  same  base  an  ordinary  white- 
light  burner  with  a  cock,  so  that  the  sodium  light  can  be  replaced 
by  white  light  by  merely  turning  the  cock. 

The  objection  to  this  form  of  sodium  burner  is  that  the  light 
which  it  furnishes  is  too  faint  for  experiments  like  those  of  the 
Fresnel. mirrors,  in  which  all  the  light  used  has  to  pass  through  a 
narrow  slit.  A  brighter  sodium  light  can  be  obtained  by  heating 
the  hard  glass  in  the  flame  of  an  oxyhydrogen  blow-pipe. 

The  most  satisfactory  light  for  these  experiments  is,  however, 
a  portion  of  the  solar  spectrum.  To  obtain  this  it  is  merely 
necessary  to  pass  the  sunlight  through  an  ordinary  spectrometer 
and  allow  the  solar  spectrum  to  fall  upon  the  slit  which  acts  as 
the  source  of  light.  A  simple  device  for  producing  such  a  spec- 
trum is  shown  in  Fig.  41.  Sunlight  passes  through  the  slit  $and 
then  through  the  upper  half  of  the  lens  L  which  renders  it  paral- 
lel. It  then  traverses  the  prism  P,  is  reflected  by  the  mirror  Jf, 
and  returns  through  the  lower  half  of  the  prism  and  the  lens,  and 
forms  a  spectrum  below  the  slit.  A  small  total  reflection  prism  p 

183 


184  APPENDIX 

turns  the  light  to  one  side  so  that  the  spectrum  is  formed  in  a 
position  convenient  for  observation.  The  advantage  of  this  form 
of  instrument  is  that  it  gives  a  rather  large  dispersion  with*  only 


FIGURE  41 


one  prism,  and  that  it  allows  the  different  parts  of  the  spectrum 
to  be  brought  upon  the  slit  of  the  instrument  being  used,  by 
merely  changing  the  position  of  the  mirror  M.  Such  a  spectro- 
scope is  easily  constructed,  as  it  does  not  require  the  careful 
adjustment  usually  necessary  in  this  kind  of  instrument. 

The  vacuum  tubes  are  useful  for  furnishing  the  cadmium  and 
mercury  light,*  their  success  depending  upon  their  having  been 
sealed  when  the  vapor  pressure  in  them  is  just  correct.  Expe- 
rience has  shown  that  the  pressure  is  correct  when  the  tube, 
excited  by  the  electric  spark,  shows  stratifications  about  1  mm. 
apart.  The  amount  of  cadmium  or  mercury  needed  is  small, 
about  the  size  of  a  pin  head. 

The  cadmium  tubes  have  to  be  heated  to  about  270°  C.  before 
the  cadmium  vaporizes.  To  make  the  heating  uniform  the  tube 
should  be  inclosed  in  a  heavy  brass  box  in  which  is  a  small  win- 
dow covered  with  mica.  The  mercury  tubes  have  to  be  warmed 
somewhat,  not  over  100°  C.  A  metal  box  is  not  necessary  with 
them,  although  it  is  desirable  to  have  one  to  keep  the  temper- 
ature fairly  uniform  throughout  the  entire  tube.  Empty  tubes 
can  be  purchased  from  any  good  glass-blower. 

Filtered    sunlight    is    sufficiently    monochromatic   for   many 

4         *  These  can  be  purchased  from  William  Gaertner,  5347  Lake  Avenue, 
Chicago,  or  can  be  readily  made. 


APPEXDIX 


185 


experiments,  notably  those  in  polarized  light.  The  following 
solutions,  recommended  by  Landolt,*  will  be  found  to  be  satisfac- 
tory: 


COLOR 

THICKNESS  or 
LAYER  IN  mm. 

AQUEOUS  SOLUTION  OF 

GRAMS  IN 
100  cc. 

AVERAGE 

A  -10« 

Red 

20 

Crystal  violet  5  BO 

0  005 

656 

Green  

20 
20 

Potassium  chromate  . 
Copper  chloride  

10 
'60 

533 

Blue  

20 

20 

Potassium  chroinate. 
Crystal  violet  

10 
0005 

448 

20 

Copper  sulphate  .... 

15 

B.   SILVERING  OP  OPTICAL  SURFACES 
The  optical  surfaces  used  in  interferometer  work  have  to  be 

coated  on  their  front  faces  with  silver.      One  of  the  simplest 

methods  of  doing  this  is  the  following: 
Prepare  two  solutions  as  follows : 


Silver  nitrate 5  gm. 

Distilled  water 40  cc. 

To  this  add  ammonia  slowly  until  the  precipitate  which  is  at 
first  formed  is  nearly  redissolved.  The  success  of  the  solution 
depends  upon  leaving  an  excess  of  the  precipitate.  If  a  drop  too 
much  ammonia  has  been  added,  a  small  crystal  of  silver  nitrate 
must  be  put  in  to  bring  back  traces  of  the  precipitate.  When 
the  solution  is  right  it  will  look  like  slightly  muddy  water.  Then 
dilute  to  500  cc.  and  filter. 


B 

Silver  nitrate 1  gm. 

Rochelle  salt  (sodium-potassium  tartrate) 0. 83  gm. 

Disti lied  water . .  . .  500  cc. 


*  Landolt,  Optische  Drehungsvermogen,  Braunschweig,  1898,  p.  390. 


186  APPENDIX 

Bring  the  solution  to  a  boil  and  filter  hot.  It  must  be  cooled 
to  the  temperature  of  the  room  before  it  is  used. 

For  silvering  use  equal  parts  of  A  and  B. 

The  essential  of  success  in  silvering  is,  besides  the  ammonia  in 
solution  A,  cleanliness.  The  following  process  of  cleaning  the 
surfaces  to  be  silvered  is  recommended : 

1.  Kemove  wax,  if  there  is  any,  with  spirits  of  turpentine. 

2.  Wash  off  the  turpentine  with  soap  and  water. 

3.  Place  the  surfaces  to  be  silvered  in  a  glass  or  porcelain 
tray,  and  remove  any  remaining  silver  with  strong  nitric  acid. 

4.  Rinse  well  in  running  water. 

5.  Wash  in  a  strong  solution  of  caustic  potash.     During  this 
washing  the  plates  and  the    dish  which  holds  them  should  be 
rubbed  hard  with  a  tuft  of  cotton  or  a  piece  of  pure  gum  tubing 
on  the  end  of  a  glass  rod.     The  success  of  the  process  depends 
largely  upon  the  thoroughness  of  this  washing. 

6.  Pour  off  the  solution  of  caustic  potash  and  rinse  well  in 
running  water,  being  careful  not  to  touch  either  the  surfaces  or 
the  inside  of  the  dish  with  the  fingers.     A  very  minute  trace  of 
grease  will  make  the  film  streaked. 

7.  Wash  in  strong  nitric  acid. 

8.  Wash  in  running  water  for  five  minutes  or  more,  raising 
the  surfaces  with  a  glass  rod  to  allow  the  water  to  run  beneath 
them. 

9.  Rinse  in  several  changes  of  distilled  water. 

Now  mix  the  two  solutions  and  pour  over  the  surfaces.  If 
only  a  thin  coat  is  desired,  the  deposit  must  be  watched,  and  the 
surface  removed  when  the  film  has  the  necessary  thickness.  The 
opaque  films  should  remain  in  the  solution  till  it  turns  black. 
The  surfaces  are  then  removed  and  set  up  on  edge  on  filter  paper 
to  dry.  When  dry  they  may  be  polished  by  rubbing  them  gently 
on  a  piece  of  chamois  skin  laid  on  the  table  and  covered  with 
jewelers'  rouge.  The  transparent  films  can  not  be  polished,  for  a 


APPENDIX 


187 


slight  touch  will  rub  the  coating  off  entirely.  If  the  solution  is 
successful  the  opaque  films  will  be  so  hard  that  they  can  not  be 
rubbed  off  with  the  finger. 

Table  1 
WAVE  LENGTHS  OF  SOME  OF  THE  IMPORTANT  LINES 


SOLAR  LINE 

SUBSTANCE 

A  •  107 

SOLAR  LIXE 

SUBSTANCE 

A  -  107 

A 

7600 

Tl. 

5350 

B 

0. 

6870 

E 

Ca. 

5270 

C 

H. 

6563 

b 

Mg. 

5173 

Cd. 

6438 

Cd. 

5086 

Di 

Na 

5896 

F 

H. 

4861 

D* 

Na. 

5890 

Cd. 

4800 

Hg. 

5790 

Hg. 

4358 

Hg.       5770 

G 

Fe.  Ca. 

4308 

Hg.       5461 

H 

H.  Ca. 

3968 

Table  2 

INDICES  OF  REFRACTION 


SOLAS  LINK 

A 

B 

C 

D 

E 

F 

G 

H 

Water 

1  3293 

3309 

3317 

3335 

3358 

3377 

3412 

.3441 

Alcohol  

1.3586 

.3599 

.3606 

.3624 

.3647 

.3667 

.3705 

.3736 

Carbon  bisulphide 
Cassia  oil   

1.6103 
1  5861 

.6166 
.5920 

.6198 
5962 

.6293 
.6053 

.6421 
.6191 

.6541 
.6340 

.6786 
.6652 

.7016 
.7010 

Crown  glass  

1.5099 

.5118 

.5127 

.5153 

.5186 

.5214 

.5267 

.5312 

Flint  glass  

1.7351 

.7406 

.7434 

.7515 

.7623 

.7723 

.7922 

.8110 

Table  3 

USEFUL  NUMBERS 

of  the  natural  system  of  logarithms  e  =  2.7183;  log  e  =  .43429 
Modulus  of  the  natural  logarithms  M=^~^=  2.3026;  log  M  =. 

Angle  whose  arc  is  equal  to  the  radius  =  57°.  2958  =  206265". 

logs  1.75812      5.31442. 
Index  of  refraction  of  air  =  1.00029. 


188 


APPENDIX 


NATURAL  SINES 


Complement 

Angle 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Difference 

0° 

0.0000 

0017 

0035 

0052 

0070 

0087 

0105 

0122 

0140 

0157 

0175 

89° 

1 

0175 

0192 

0209 

0227 

0244 

0262 

0279 

0297 

0314 

0332 

0349 

88 

2 

0349 

0366 

0384 

0401 

0419 

0436 

0454 

0471 

0488 

0506 

0523 

87 

8 

0523 

0541 

0558 

0576 

0593 

0610 

0628 

0645 

0663 

0680 

0698 

86 

4 

0698 

0715 

0732 

0750 

0767 

0785 

0802 

0819 

0837 

0854 

0872 

85 

5 

0.0872 

0889 

0906 

0924 

0941 

0958 

0976 

0993 

1011 

1028 

1045 

84 

6 

1045 

1063 

1080 

1097 

1115 

1132 

1149 

1167 

1184 

1201 

1219 

83 

7 

1219 

1236 

1253 

1271 

1288 

1305 

1323 

1340 

1357 

1374 

1392 

82 

8 

1392 

1409 

1426 

1444 

1461 

1478 

1495 

1513 

1530 

1547 

1564 

81 

9 

1564 

1582 

1599 

1616 

1633 

1650 

1668 

1685 

1702 

1719 

1736 

80 

10 

0.1736 

1754 

1771 

1788 

1805 

1822 

1840 

1857 

1874 

1891 

1908 

79 

11 

1908 

1925 

1942 

1959 

1977 

1994 

2011 

2028 

2045 

2062 

2079 

78 

12 

2079 

2096 

2113 

2130 

2147 

2164 

2181 

2198 

2215 

2233 

2250 

77  17 

13 

2250 

2267 

2284 

2300 

2317 

2334 

2351 

2368 

2385 

2402 

2419 

76 

14 

2419 

2436 

2453 

2470 

2487 

2504 

2521 

2538 

2554 

2571 

2588 

75 

15 

0.2588 

2605 

2622 

2639 

2656 

2672 

2689 

2706 

2723 

2740 

2756 

74 

16 

2756 

2773 

2790 

2807 

2823 

2840 

2857 

2874 

2890 

2907 

2924 

73 

17 

2924 

2940 

2957 

2974 

2990 

3007 

3024 

3040 

3057 

3074 

3090 

72 

18 

3090 

3107 

3123 

3140 

3156 

3173 

3190 

3206 

3223 

3239 

3256 

71 

19 

3256 

3272 

3289 

3305 

3322 

3338 

3355 

3371 

3387 

3404 

3420 

70 

20 

0.3420 

3437 

3453 

3469 

3486 

3502 

3518 

3535 

3551 

3567 

3584 

69 

21 

3584 

3600 

3616 

3633 

3649 

3665 

3681 

3697 

3714 

3730 

3746 

68 

22 

3746 

3762 

3778 

3795 

3811 

3827 

3843 

3859 

3875 

3891 

3907 

67 

23 

3907 

3923 

3939 

3955 

3971 

3987 

4003 

4019 

4035 

4051 

4067 

66  16 

24 

4067 

4083 

4099 

4115 

4131 

4147 

4163 

4179 

4195 

4210 

4226 

65 

25 

0.4226 

4242 

4258 

4274 

4289 

4305 

4321 

4337 

4352 

4368 

4384 

64 

26 

4384 

4399 

4415 

4431 

4446 

4462 

4478 

4493 

4509 

4524 

4540 

63 

27 

4540 

4555 

4571 

4586 

4602 

4617 

4633 

4648 

4664 

4679 

4695 

62 

28 

4695 

4710 

4726 

4741 

4756 

4772 

4787 

4802 

4818 

4833 

4848 

61 

29 

4848 

4863 

4879 

4894 

4909 

4924 

4939 

4955 

4970 

4985 

5000 

60 

30 

0.5000 

5015 

5030 

5045 

5060 

5075 

5090 

5105 

5120 

5135 

5150 

59  15 

31 

5150 

5165 

5180 

5195 

5210 

5225 

5240 

5255 

5270 

5284 

5299 

58 

32 

5299 

5314 

5329 

5344 

5358 

5373 

5388 

5402 

5417 

5432 

5446 

57 

33 

5446 

5461 

5476 

5490 

5505 

5519 

5534 

5548 

5563 

5577 

5592 

56 

34 

5592 

5606 

5621 

5635 

5650 

5664 

5678 

5693 

5707 

5721 

5736 

55 

35 

0.5736 

5750 

5764 

5779 

5793 

5807 

5821 

5835 

5850 

5864 

5878 

54 

36 

5878 

5892 

5906 

5920 

5934 

5948 

5962 

5976 

5990 

6004 

6018 

63  » 

37 

6018 

6032 

6046 

6060 

6074 

6088 

6101 

6115 

6129 

6143 

6157 

52 

38 

6157 

6170 

6184 

6198 

6211 

6225 

6239 

6252 

6266 

6280 

6293 

51 

39 

6293 

6307 

6320 

6334 

6347 

6361 

6374 

6388 

6401 

6414 

6428 

50 

40 

0.6428 

6441 

6455 

6468 

6481 

6494 

6508 

6521 

6534 

6547 

6561 

49 

41 

6561 

6574 

6587 

6600 

6613 

6626 

6639 

6652 

6665 

6678 

6691 

48  13 

42 

6691 

6704 

6717 

6730 

6743 

6756 

6769 

6782 

6794 

6807 

6820 

47 

43 

6820 

6833 

6845 

6658 

6871 

6884 

6896 

6909 

6921 

6934 

6947 

46 

44° 

6947 

6959 

6972 

6984 

6997 

7009 

7022 

7034 

7046 

7059 

7071 

45° 

Complement 

.9 

.8 

.7 

.6 

.5 

.4 

.3 

.2 

.1 

.0 

Angle 

NATURAL  COSINES 


APPENDIX 


189 


NATURAL  SINES 


Angle 

.0 

.1 

.2 

.3 

A 

.5 

.6 

.7 

.8 

.9 

Complement 
Difference 

45° 

0.7071 

7083 

7096 

7108 

7120 

7133 

7145 

7157 

7169 

7181 

7193 

44° 

46 

7193 

7206 

7218 

7230 

7242 

7254 

7266 

7278 

7290 

7302 

7314 

43  12 

47 

7314 

7325 

7337 

7349 

7361 

7373 

7385 

7396 

7408 

7420 

7431 

42 

48 

7431 

7443 

7455 

7466 

7478 

7490 

7501 

7513 

7524 

7536 

7547 

41 

49 

7547 

7559 

7570 

7581 

7593 

7604 

7615 

7627 

7638 

7649 

7660 

40 

50 

0.7660 

7672 

7683 

7694 

7705 

7716 

7727 

7738 

7749 

7760 

7771 

39 

51 

7771 

7782 

7793 

7804 

7815 

7826 

7837 

7848 

7859 

7869 

7880 

38  » 

52 

7880 

7891 

7902 

7912 

7923 

7934 

7944 

7955 

7965 

7976 

7986 

37 

53 

7986 

7997 

8007 

8018 

8028 

8039 

8049 

8059 

8070 

8080 

8090 

36 

54 

8090 

8100 

8111 

8121 

8131 

8141 

8151 

8161 

8171 

8181 

8192 

35 

55 

0.8192 

8202 

8211 

8221 

8231 

8241 

8251 

8261 

8271 

8281 

8290 

34  * 

56 

8290 

8300 

8310 

8320 

8329 

8339 

8348 

8358 

8368 

8377 

8387 

33 

57 

8387 

8396 

8406 

8415 

8425 

8434 

8443 

8453 

8462 

8471 

8480 

32 

58 

8480 

8490 

8499 

8508 

8517 

8526 

8536 

8545 

8554  8563 

8572 

31 

59 

8572 

8581 

8590 

8599 

8607 

8616 

8625 

8634 

8643 

8652 

8660 

30  9 

60 

0.8660 

8669 

8678 

8686 

8695 

8704 

8712 

8721 

8729 

8738 

8746 

29 

61 

8746 

8755 

8763 

8771 

8780 

8788 

8796 

8805 

8818 

8821 

8829 

28 

62 

8829 

8838 

8846 

8854 

s*02 

8870 

8878 

8886  8894 

8902 

8910 

27  8 

63 

8910 

8918 

8926 

8934 

8942 

8949 

8957 

8965 

8973 

8980 

8988 

26 

64 

8988 

8996 

9003 

9011 

9018 

9026 

9033 

9041 

9048 

9056 

9063 

25 

65 

0.9063 

9070 

9078 

9085 

9092 

9100 

9107 

9114 

9121 

9128 

9135 

24 

.66 

9135 

9143 

9150 

9157 

9164 

9171 

9178 

9184 

9191 

9198 

9205 

23  * 

67 

9205 

9212 

9219 

9225 

9232 

9239 

9245 

9252 

9259 

9265 

9272 

22 

68 

9272 

9278 

9285 

9291 

9298 

9304 

9311 

9317 

9323 

9330 

9336 

21 

69 

9336 

9342 

9348 

9354 

9361 

9367 

9373 

9379 

9385 

9391 

9397 

20  6 

70 

0.9397 

9403 

9409 

9415 

9421 

9426 

9432 

9438 

9444 

9449 

9455 

19 

71 

9455 

9461 

9466 

9472 

9478 

9483 

9489 

9494 

9500 

9505 

9511 

18 

72 

9511 

9516 

9521 

9527 

9532 

9537 

9542 

9548 

9553 

9558 

9563 

17 

73 

9563 

9568 

9573 

9578 

9583 

9588 

9593 

9598 

9603 

9608 

9613 

16  5 

74 

9613 

9617 

9622 

9627 

9632 

9636 

9641 

9646 

9650 

9655 

9659 

15 

75 

0.9659 

9664 

9668 

9673 

9677 

9681 

9686 

9690 

9694 

9699 

9703 

14 

76 

9703 

9707 

9711 

9715 

9720 

9724 

9728 

9732 

9736 

9740 

9744 

13  * 

77 

9744 

9748 

9751 

9755 

9759 

9763 

9767 

9770 

9774 

9778 

9781 

12 

78 

9781 

9785 

9789 

9792 

9796 

9799 

9803 

9806 

9810 

9813 

9816 

11 

79 

9816 

9820 

9823 

9826 

9829 

9833 

9836 

9839 

9842 

9845 

9848 

10 

80 

0.9848 

9851 

9854 

9857 

9860 

9863 

9866 

9869 

9871 

9874 

9877 

9  3 

81 

9877 

9880 

uss2 

9885 

9888 

9890 

9893 

9895 

9898 

9900 

9903 

8 

82 

9903 

9905 

9907 

9910 

9912 

9914 

9917 

9919 

9921 

9923 

9925 

7 

83 

9925 

9928 

9930 

9932 

9934 

9936 

9938 

9940 

9942 

9943 

9945 

6  2 

84 

9945 

9947 

9949 

9951 

9952 

9954 

9956 

9957 

9959 

9960 

9962 

5 

85 

0.9962 

9963 

9965 

9966 

9968 

9969 

9971 

9972 

9973 

9974 

9976 

4 

86 

9976 

9977 

9978 

9979 

9980 

9981 

9982 

9983 

9984 

9985 

9986 

3  i 

87 

9986 

9987 

9988 

9989 

9990 

9990 

9991 

9992 

9993 

9993 

9994 

2 

88 

9994 

9995  9995 

9996 

9996 

9997 

9997 

9997 

9998 

9998 

9998 

1 

89° 

9998 

9999  9999 

9999 

9999 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

0°° 

Complement 

.9 

.8 

.7 

.6 

.5 

.4 

.3 

.2 

.1 

.0 

Angle 

NATURAL  COSINES 


190 


APPENDIX 


NATURAL  TANGENTS 


Com  plement 

Angle 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Difference 

0 

0.0000 

0017 

0035 

0052 

0070 

0087 

0105 

0122 

0140 

0157 

0175 

89° 

1 

0175 

0192 

0209 

0227 

0244 

0262 

0279 

0297 

0314 

0332 

0349 

88 

2 

0349 

0367 

0384 

0402 

0419 

0437 

0454 

0472 

0489 

0507 

0524 

87 

3 

0524 

0542 

0559 

0577 

0594 

0612 

0629 

0647 

0664 

0682 

0699 

86 

4 

0699 

0717 

0734 

0752 

0769 

0787 

0805 

0822 

0840 

0857 

0875 

85 

5 

0.0875 

0892 

0910 

0928 

0945 

0963 

0981 

0998 

1016 

1033 

1051 

84 

6 

1051 

1069 

1086 

1104 

1122 

1139 

1157 

1175 

1192 

1210 

1228 

83 

7 

1228 

1246 

1263 

1281 

1299 

1317 

1334 

1352 

1370 

1388 

1405 

82 

8 

1405 

1423 

1441 

1459 

1477 

1495 

1512 

1530 

1548 

1566 

1584 

81 

9 

1584 

1602 

1620 

1638 

1655 

1673 

1691 

1709 

1727 

1745 

1763 

80 

10 

0.1763 

1781 

1799 

1817 

1835 

1853 

1871 

1890 

1908 

1926 

1944 

79  M 

11 

1944 

1962 

1980 

1998 

2016 

2035 

2053 

2071 

2089 

2107 

2126 

78 

12 

2126 

2144 

2162 

2180 

2199 

2217 

2235 

2254 

2272 

2290 

2309 

77 

13 

2309 

2327 

2345 

2364 

2382 

2401 

2419 

2438 

2456 

2475 

2493 

76 

14 

2493 

2512 

2530 

2549 

2568 

2586 

2605 

2623 

2642 

2661 

2679 

75 

15 

0.2679 

2698 

2717 

2736 

2754 

2774 

2792 

2811 

2830 

2849 

2867 

74 

16 

2867 

2886 

2905 

2924 

2943 

2962 

2981 

3000 

3019 

3038 

3057 

73  19 

17 

3057 

3076 

3096 

3115 

3134 

3153 

3172 

3191 

3211 

3230 

3249 

72 

18 

3249 

3269 

3288 

3307 

3327 

3346 

3365 

3385 

3404 

3424 

3443 

71 

19 

3443 

3463 

3482 

3502 

3522 

3541 

3561 

3581 

3600 

3620 

3640 

70 

20 

0.3640 

3659 

3679 

3699 

3719 

3739 

3759 

3779 

3799 

3819 

3839 

69 

21 

3839 

3859 

3879 

3899 

3919 

3939 

3959 

3979 

4000 

4020 

4040 

68  20 

22 

4040 

4061 

4081 

4101 

4122 

4142 

4163 

4183 

4204 

4224 

4245 

67 

23 

4245 

4265 

4286 

4307 

4327 

4348 

4369 

4390 

4411 

4431 

4452 

66 

24 

4452 

4473 

4494 

4515 

4536 

4557 

4578 

4599 

4621 

4642 

4663 

65  21 

25 

0.4663 

4684 

4706 

4727 

4748 

4770 

4791 

4813 

4834 

4856 

4877 

64 

26 

4877 

4899 

4921 

4942 

4964 

4986 

5008 

5029 

5051 

5073 

5095 

63 

27 

5095 

5117 

5139 

5161 

5184 

5206 

5228 

5250 

5272 

5295 

5317 

62  22 

28 

5317 

5340 

5362 

5384 

5407 

5430 

5452 

5475 

5498 

5520 

5543 

61 

29 

5543 

5566 

5589 

5612 

5635 

5658 

5681 

5704 

5727 

5750 

5774 

60  23 

30 

0.5774 

5797 

5820 

5844 

5867 

5890 

5914 

5938 

5961 

5985 

6099 

59 

31 

6009 

6032 

6056 

6080 

6104 

6128 

6152 

6176 

6200 

6224 

6249 

58  2* 

32 

6249 

6273 

6297 

6322 

6346 

6371 

6395 

6420 

6445 

6469 

6494 

57 

33 

6494 

6519 

6544 

6569 

6594 

6619 

6644 

6669 

6694 

6720 

6745 

56  25 

34 

6745 

6771 

6796 

6822 

6847 

6873 

6899 

6924 

6950 

6976 

7002 

55 

35 

0.7002 

7028 

7054 

7080 

7107 

7133 

7159 

7186 

7212 

7239 

7265 

5426 

36 

7265 

7292 

7319 

7346 

7373 

7400 

7427 

7454 

7481 

7508 

7536 

53  * 

37 

7536 

7563 

7590 

7618 

7646 

7673 

7701 

7729 

7757 

7785 

7813 

52  28 

38 

7813 

7841 

7869 

7898 

7926 

7954 

7983 

8012 

8040 

8069 

8098 

51  28 

39 

8098 

8127 

8156 

8185 

8214 

8243 

8273 

8302 

8332 

8361 

8391 

50  29 

40 

0.8391 

8421 

8451 

8481 

8511 

8541 

8571 

8601 

8632 

8662 

8693 

49  30 

41 

8693 

8724 

8754 

8785 

8816 

8847 

8878 

8910 

8941 

8972 

9004 

48  31 

42 

9004 

9036 

9067 

9099 

9131 

9163 

9195 

9228 

9260 

9293 

9325 

47  32 

43 

9325 

9358 

9391 

9424 

9557 

9490 

9523 

9556 

9590 

9623 

9657 

46  33 

44° 

9657 

9691 

9725 

9759 

9793 

9827 

9861 

9896 

9930 

9965 

1.0000 

45034 

Complement 

.9 

.8 

.7 

.6 

.5 

.4 

.3 

.2 

•l 

.0 

Angle 

NATURAL  COTANGENTS 


APPENDIX 

NATURAL  TANGENTS 


191 


Angle]      .0 

.1 

.2 

.3 

A 

.5 

.6 

.7 

.8 

.9 

Dif. 

45 

1.0000 

1.0035  1.0070 

1.0105 

1.0141 

1.0176 

1.0212 

1.0247 

L.0388 

1.0319 

36 

46 

1.0355 

1.0392 

1.0428 

1.0464 

1.0501 

1.0538 

1.0575 

1.0612 

1.0649 

1.0686 

• 

47 

1.07-24 

1.0761 

1.0799 

L.063-3 

L087B 

1.0913 

1.0951 

1.0990 

1.1028 

1.1067 

38 

48 

1.1106 

1.11451.1184 

1.1224 

1.1263 

1.1303 

1.1343 

1.1383 

1.1423 

1.1463 

• 

49 

1.1504 

1.1544 

1.1585 

1.1626 

1.1667 

1.1708 

1.1750 

1.1792 

1.1833 

1.1875j    « 

50 

1.1918 

1.1960 

1.2002 

1.2045 

L2088 

1.2131 

1.2174 

1.2218 

1.2261 

1.2305    « 

51 

.23491.23931.2437 

1.2482 

1.2527 

1.2572 

1.2617 

1.2662 

1.2708 

1.2753    « 

52 

.2799  1.2846|1.2892 

1.2985J1.3032 

1.3079 

1.3127 

1.3175  1.3222     *T 

58 

.3270  1.3319  1.3367 

1.3416  1.3465  1.3514 

1.3564 

1.3613 

1.36631.3713     « 

54 

.3764 

1.3814 

1.3865|1.3916 

1.3968 

1.4019 

1.4071 

1.4124 

1.4176 

1.4229 

53 

55 

.4281 

1.4335 

1.43881.4442 

1.4496 

1.4550 

1.4605 

1.4659 

1.4715 

1.4770 

54 

56 

.4826|l.48£                ..4994  l.oo.M  1.510s  1.5166 

1.5224 

1.5282 

1.5340    si 

57 

.5399  1.5458 

1.5517  1.5577  1.5637  1.5697  1.5757 

1.5818 

L0880 

1.5941'    » 

58 

.60031.6066 

1.6128  1.6191  1.6255  1.6319  1.6383 

1.6447 

1.6512 

1.6577     « 

59 

.66431.6709 

1.6775 

1.6842 

1.6909 

1.6977 

1.7045 

1.7113 

1.7182 

1.7251 

68 

60 

.732l'l.739l 

1.7461 

1.7532 

1.7603 

1.7675 

1.7747 

1.7820 

1.7893 

1.7966 

72 

61 

.8040 

1.8115 

1.8190 

1.8265 

1.83411.84181.8495 

1.8572  1.8650 

1.872*     " 

62 

.  —  •-. 

:.  —  7 

1.8967 

1.9047 

1.9128  1.9210  1.9292 

1.9375  1.9458 

1.9542    82 

68 

.9626 

1.9711 

1.9797 

L9669 

1.  9970  2.  0057  2.  0145  2.  0233  2.  0323 

2.0413    88 

64 

2.0503 

2.0594 

2.0686 

2.0778 

2.0872 

2.0965 

2.1060 

2.11552.1251 

2.1348    * 

65 

2.145 

2.154 

2.164 

2.174 

2.184 

2.194 

2.204 

2.215  '2.225 

2.236 

10 

66 

2.246   2.-2.J7 

2.267 

2.278 

2.289 

2.300 

2.311 

2.322    2.333    2.344 

11 

67 

2.&56 

2.367 

2.379 

2.391 

2.402 

2.414 

2.426 

2.438   2.450   2.463 

:. 

68 

2.475 

2.488 

2.500 

2.513 

2.526 

2.539 

2.552 

2,565    2.  .57*    2.592 

13 

69 

2.605 

2.619 

2.633 

2.646 

2.660 

2.675 

&680 

2.703 

2.718 

2.733 

U 

70 

2.747 

2.762 

•2.77<3 

2.793 

1806 

2.824 

2.840 

2,856 

2.872 

2.888 

16 

71 

2.904 

2.921 

2.937 

2.954   2.971 

2.989 

3.006  13.024   3.042 

3.060 

n 

72 

3.078 

3.096 

3:115 

3.133   3.152 

3.172 

3.191 

3.211    3.230 

3.250 

19 

78 

3.271 

3.291 

3.312 

3.333 

3.354 

3.376 

3.398 

3.420 

3.442 

3.465 

74 

3.487 

3.511 

3.534 

3.558 

3.582 

3.606 

3.630 

3.655 

3.681 

3.700 

» 

75 

1.782 

3.758 

3.785 

3.812 

3.839 

3.867 

3.895 

3.923 

3.952 

3.981 

28 

76 

4.011 

4041 

4071 

4102 

4134 

4165 

4198 

4230 

4264 

4297 

32 

77 

4.331 

4366 

4402 

4437 

4474 

4511 

4548 

4586 

4625 

4.665 

37 

78 

4705 

4745 

4.7-7 

4829 

4872 

4.915 

4959 

5.005 

5.050 

5.097 

44 

79 

5.145 

5.193 

5.243 

5.292 

5.343 

5.396 

5.449 

5.503 

5.558 

5.614 

52 

80 

5.67 

5.73 

5.79 

5.85 

5.91 

5.98 

6.04 

6.11 

6.17 

6.24 

T 

81 

6.31 

6.39 

6.46 

6.54 

6.61 

6.69 

6.77 

6.85 

6.94 

7.03 

8 

82 

7.12 

7.21 

7.30 

7.40 

7.49 

7.60 

7.70 

7.81 

7.92 

8.03 

10 

88 

8.14 

8.26 

8.39 

8.51 

8.64 

8.78 

8.92 

9.06 

9.21 

9.36 

14 

84 

9.51 

9.68 

9.84 

10.0 

10.2 

10.4 

10.6 

10.8 

11.0 

11.2 

85 

11.4 

11.7 

11.9 

12.2 

12.4 

12.7 

13.0 

13.3 

13.6 

140 

* 

86 

14.3 

147 

15.1 

15.5 

15.9 

16.3 

16.8 

17.3 

17.9 

18.5 

6 

87 

19.1 

19.7 

20.4 

21.2 

22.0 

22.9 

23.9 

249 

26.0 

27.3 

88 

28.6 

30.1 

31.8 

33.7 

35.8 

38.2 

40.9 

44.1 

47.7 

52.1 

89 

57. 

64. 

72. 

v2. 

95. 

115. 

143. 

191. 

286. 

573. 

Angle 

.0 

.1 

9 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

NATURAL  TANGENTS 


192 


APPENDIX 


LOGARITHMS 


o 

l 

2 

3 

4  . 

5 

6 

'  7 

8 

9 

1  2  3 

456 

789 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

4  8  12 

17  21  25 

29  33  37 

11 

12 
13 

0414 
0792 
1139 

0453 
0828 
1173 

0492 
0864 
1206 

0531 
0899 
1239 

0569 
0934 
1271 

0607 
0969 
1303 

0645 
1004 
1335 

0682 
1038 
1367 

0719 
1072 
1399 

0755 
1106 
1430 

4  8  11 
3  7  10 
3  6  10 

15  19  23 
14  17  21 
13  16  19 

26  30  34 
24  28  31 
23  26  29 

14 
15 
16 

1461 
1761 
2041 

1492 
1790 
2068 

1523 

1818 
2095 

1553 
1847 
2122 

1584 
1875 
2148 

1614 
1903 
2175 

1644 
1931 
2201 

1673 
1959 

2227 

1703 
1987 
2253 

1732 
2014 
2279 

369 

368 
358 

12  15  18 
11  14  17 
11  13  16 

21  24  27 
20  22  25 
18  21  24 

17 

18 
19 

2304 
2553 

2788 

2330 

2577 
2810 

2355 
2601 
2833 

2380 
2625 

2856 

2405 
2648 

2878 

2430 
2672 
2900 

2455 
2695 
2923 

2480 
2718 
2945 

2504 

2742 
2967 

2529 
2765 
2989 

257 
257 
247 

10  12  15 
9  12  14 
9  11  13 

17  20  22 
16  19  21 
16  18  20 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

246 

8  11  13 

15  17  19 

21 
22 
23 

3222 
3424 
3617 

3243 
3444 
3636 

3263 
3464 
3655 

3284 
3483 
3674 

3304 
3502 
3692 

3324 
3522 
3711 

3345 
3541 
3729 

3365 
3560 

3747 

3385 
3579 
3766 

3404 
3598 

3784 

246 
246 
246 

8  10  12 
8  10  12 
7  9  11 

14  16  18 
14  15  17 
13  15  17 

24 
25 
26 

3802 
3979 
4150 

3820 
3997 
4166 

3838 
4014 
4183 

3856 
4031 
4200 

3874 
4048 
4216 

3892 
4065 
4232 

3909 
4082 
4249 

3927 
4099 
4265 

3945 
4116 
4281 

3962 
4133 

4298 

245 
235 
235 

7  9  11 
7  9  10 

7  8  10 

12  14  16 
12  14  15 
11  13  15 

27 

28 
29 

4314 

4472 
4624 

4330 

4487 
4639 

4346 
4502 
4654 

4362 
4518 
4669 

4378 
4533 
4683 

4393 

4548 
4698 

4409 
4564 
4713 

4425 
4579 

4728 

4440 
4594 

4742 

4456 
4609 

4757 

235 
235 
1  3  4 

689 
689 
679 

11  13  14 
11  12  14 
10  12  13 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

1  3  4 

679 

10  11  13 

31 
32 
33 

4914 
5051 

5185 

4928 
5065 
5198 

4942 
5079 
5211 

4955 
5092 
5224 

4969 
5105 
5237 

4983 
5119 
5250 

4997 
5132 
5263 

5011 
5145 
5276 

5024 
5159 
5289 

5038 
5172 
5302 

1  8  4 
1  3  4 
1  3  4 

678 
578 
568 

10  11  12 
9  11  12 
9  10  12 

34 
35 
86 

5315 
5441 
5563 

5328 
5453 
5575 

5340 
5465 

5587 

5353 

5478 
5599 

5366 
5490 
5611 

5378 
5502 
5623 

5391 
5514 
5635 

5403 
5527 
5647 

5416 
5539 
5658 

5428 
5551 
5670 

1  3  4 
1  2  4 
1  2  4 

568 
567 
567 

9  10  11 
9  10  11 
8  10  11 

37 

38 
39 

5682 
5798 
5911 

5694 
5809 
5922 

5705 
5821 
5933 

5717 
5832 
5944 

5729 
5843 
5955 

5740 
5855 
5966 

5752 

5866 
5977 

5763 

5877 
5988 

5775 
5888 
5999 

5786 
5899 
6010 

1  2  3 
1  2  3 
1  2  3 

567 
567 

457 

8  9  10 
8  9  10 
8  9  10 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

1  2  3 

456 

8  9  10 

41 
42 
43 

6128 
6232 
6335 

6138 
6243 
6345 

6149 
6253 
6355 

6160 
6263 
6365 

6170 
6274 
6375 

6180 
6284 
6385 

6191 
6294 
6395 

6201 
6304 
6405 

6212 
6314 
6415 

6222 
6325 
6425 

1  2  3 
123 
123 

456 
456 
456 

789 
789 
789 

44 
45 
46 

6435 
6532 
6628 

6444 
6542 
6637 

6454 
6551 
6646 

6464 
6561 
6656 

6474 
6571 
6665 

6484 
6580 
6675 

6493 
6590 
6684 

6503 
6599 
6693 

6513 
6609 
6702 

6522 

6618 
6712 

1  2  3 
123 
1  2  3 

456 
456 
456 

789 
789 
778 

47 

48 
49 

6721 
6812 
6902 

6730 
6821 
6911 

6739 
6830 
6920 

6749 
6839 
6928 

6758 
6848 
6937 

6767 

6857 
6946 

6776 
6866 
6955 

6785 
6875 
6964 

6794 
6884 
6972 

6803 
6893 
6981 

123 
123 
1  2  3 

455 
445 
445 

678 
678 
678 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

1  2  3 

345 

678 

51 
52 
53 

7076 
7160 
7243 

7084 
7168 
7251 

7093 

7177 
7259 

7101 

7185 

7267 

7110 
7193 

7275 

7118 
7202 

7284 

7126 
7210 
7292 

7135 

7218 
7300 

7143 
7226 

7308 

7152 
7235 
7316 

123 
122 

1  2  2 

345 
345 
345 

678 
677 
667 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

1  2  2 

345 

667 

APPENDIX 


193 


LOGARITHMS 


55 

0 

l 

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8  . 

9 

1  2  3 

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7459 

7466 

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122 

345 

567 

56 
57 

58 

7482 
7559 
7634 

7490 
7566 
7642 

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7664 

7520 
7597 
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7536 
7612 
7686 

7543 
7619 
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7551 
7627 
7701 

122 
122 
112 

345 
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344 

567 
567 
567 

59 
60 
61 

7709 

7782 
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7716 

7789 
7860 

7723 
7796 

7868 

7731 

7803 
7875 

7738 
7810 
7882 

7745 

7818 
7889 

7752 

7825 
7896 

7760 
7832 
7903 

7767 
7839 
7910 

7774 
7846 
7917 

112 
112 
112 

344 
344 
344 

567 
566 
566 

62 

<i:J 
64 

7924 
7993 
8062 

7931 
8000 
8069 

7938 
8007 
8075 

7945 
8014 

8082 

7952 
8021 
8089 

7959 
8028 
8096 

7966 
8035 
8102 

7973 
8041 
8109 

7980 
8048 
8116 

7987 
8055 
'8122 

112 
112 
112 

334 
334 
334 

566 
556 
556 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

1  1 

334 

556 

66 
67 

68 

8195 
8261 
8325 

8202 
8267 
8331 

8209 
8274 

8338 

82i5 
8280 
8344 

8222 

8287 
8351 

8228 
8293 
8357 

8235 
8299 
8363 

8241 
8306 
8370 

8248 
8312 
8376 

8254 
8319 
8382 

1  1 
1  1 
1  1 

334 
334 
334 

556 
556 
456 

69 
70 
71 

8388 
8451 
8513 

8395 

8457 
8519 

8401 
8463 
8525 

8407 
8470 
8531 

8414 
8476 
8537 

8420 
8482 
8543 

8426 

8488 
8549 

8432 
8494 
8555 

8439 
8500 
8561 

8445 
8506 
8567 

1  1 
1  1 
1  1 

234 
234 
234 

456 
456 
455 

72 
73 
74 

8573 
8633 
8692 

8579 
8639 
8698 

8585 
8645 
8704 

8591 
8651 
8710 

8597 
8657 
8716 

8603 
8663 
8722 

8609 
8669 

8727 

8615 
8675 
8733 

8621 
8681 
8739 

8627 
8686 
8745 

1 
1 

1 

234 
234 
234 

455 
455 
455 

::> 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

1 

233 

455 

76 

77 
78 

8808 
8865 
8921 

8814 
8871 
8927 

8820 
8876 
8932 

8825 
8882 
8938 

8831 
8887 
8943 

8837 
8893 
8949 

8842 
8899 
8954 

8848 
8904 
8960 

8854 
8910 
8965 

8859 
8915 
8971 

1 
1 
1 

233 
233 
233 

455 
445 
445 

79 

80 
81 

8976 
9031 
9085 

8982 
9036 
9090 

8987 
9042 
9096 

8993 
9047 
9101 

8998 
9053 
9106 

9004 
9058 
9112 

9009 
9063 
9117 

9015 
9069 
9122 

9020 
9074 
9128 

9025 
9079 
9133 

1 
1 
1 

233 
233 
233 

445 
445 
445 

82 
s:i 
84 

9138 
9191 
9243 

9143 
9196 
9248 

9149 
9201 
9253 

9154 
9206 
9258 

9159 
9212 
9263 

9165 
9217 
9269 

9170 
9222 
9274 

9175 
9227 
9279 

9180 
9232 
9284 

9186 
9238 
9289 

1 
1 

1 

238 
233 
233 

445 
445 
445 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

1  1 

233 

445 

86 
87 

88 

9345 
9395 
9445 

9350 
9400 
9450 

9355 
9405 
9455 

9360 
9410 
9460 

9365 
9415 
9465 

9370 
9420 
9469 

9375 
9425 
9474 

9380 
9430 
9479 

9385 
9435 

9484 

9390 
9440 
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1  1 
0  1  1 
0  1  1 

233 
223 
223 

445 
344 
344 

89 
90 
91 

9494 
9542 
9590 

9499 
9547 
9595 

9504 
9552 
9600 

9509 
9557 
9605 

9513 
9562 
9609 

9518 
9566 
9614 

9523 
9571 
9619 

9528 
9576 
9624 

9533 
9581 
9628 

9538 
9586 
9633 

0  1  1 
0  1  1 
0  1  1 

223 
223 
223 

344 
344 
344 

92 
93 
94 

9638 
9685 
9731 

9643 
9689 
9736 

9647 
9694 
9741 

9652 
9699 
9745 

9657 
9703 
9750 

9661 
9708 
9754 

9666 
9713 
9759 

9671 
9717 
9763 

9675 
9722 

9768 

9680 
9727 
9773 

0  1  1 
0  1  1 
0  1  1 

223 
223 
223 

344 
344 
344 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

Oil 

223 

344 

96 
97 
98 

9823 

9868 
9912 

9827 
9872 
9917 

9832 

9877 
9921 

9836 
9881 
9926 

9841 
9886 
9930 

9845 
9890 
9934 

9850 
9894 
9939 

9854 
9899 
9943 

9859 
9903 
9948 

9863 
9908 
9952 

0  1  1 
0  1  1 
0  1  1 

223 
223 
223 

344 
344 
344 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

0  1  1 

223 

334 

INDEX 


Abbe,  total  reflection,  107. 
Absorption,  coefficient  of,  157. 

relation  to  anomalous  dispersion, 

173. 

Achromatic  fringe,  37. 
Ames,  concave  grating,  121. 
Angle  of  incidence,  84. 

of  emergence,  84. 

principal,  157. 

Angstrom,  absolute  wave  length,  124. 
Azimuth,  of  plane  of  polarization, 
152. 

principal,  157. 
Babinet,  compensator,  143. 
Bell,  absolute  wave  length,  124. 
Biot,  rotary  polarization,  131. 
Bi-prism,  Fresnel,  44. 
Boltzmann,  rotary  dispersion,  131. 
Brewster,  law,  151. 
Cadmium,    distribution    in    spectral 

lines  of,  79. 

Cathode  rays,  175,  181. 
Cauchy,  dispersion  equation,  39,  90. 
Concave  grating,  116. 

constant  of,  121. 
Congruent  rays,  21. 
Conroy,  Brewster 's  law,  151. 

reflection  of  polarized  light,  152. 
Corpuscular  theory,  166. 
Czapski,  total  reflection,  107. 
Deviation,  minimum,  86. 
Dielectric  constant,  equals  square  of 

index,  173. 
Diffraction,  by  one  slit,  11. 

by  two  slits,  19. 

grating,  108. 

Dispersion,   Cauchy's   equation  for, 
39,  90,  173. 


Dispersion,  determination  with  the 
interferometer,  61. 

of  prism,  94. 

anomalous,  173. 

electromagnetic  theory  of,  175. 
Distribution  in  source,  determination 

by  visibility,  25,  71. 
Double  slit,  19,  23. 
Drude,  resolving  power  of  prisms,  91, 

diffraction  gratings,  108. 

reflection  of  polarized  light,  150. 

metallic  reflection,  157. 

electromagnetic  theory,  170,  172, 

174,  175,  181. 
Electrolysis,  175,  181. 
Electromagnetic,  theory,  170. 
Electron,  182. 
Elliptically  polarized  light,  140. 

analysis  of,  142. 

Fabry,  distribution  in  sources,  82. 
Faraday,     electromagnetic     theory, 

169,  178. 

Fraunhofer,  diffraction  phenomena, 
11. 

diffraction  grating,  17. 
Fresnel,  mirrors,  30,  33. 

bi-prism,  44. 

reflection  equations,  150. 
Glan,  spectrophotometer,  159. 
Glazebrook,  optical  theories,  169. 
Grating,  plane,  108. 

dispersion  of,  110. 

resolving  power  of,  111. 

constant  of,  112. 

concave,  116. 

Gubbe,  rotary  polarization,  134. 
Gumlich,  rotary  polarization,  131. 
Helmholtz,  prisms,  96. 


194 


IXDEX 


195 


Interference,  general  discussion  of, 

21. 
Interferometer,  definition  of,  32. 

Michelson,  48. 

adjustment  of,  55. 
Ions,  175. 

ratio  of  charge  to  mass,  181. 
Jamin,  Brewster's  law,  151. 
Kayser,  prisms,  96. 

plane  grating,  108. 

concave  grating,  116. 

absolute  wave  lengths,  125. 

regularities  in  spectra,  177. 
Kohlrausch,  total  reflection,  107. 
Kurlbaum,    absolute    wave   length, 

124. 

Landolt,  rotary  polarization,  133. 
Lloyd,  optical  theories,  169. 
Magnetism,  action  on  light,  178. 
Mascart,  visibility  curves,  27. 

gratings,  108. 

Maxwell,    law     of     distribution     of 
velocities  of  molecules,  75. 

electromagnetic  theory,  170,  172. 
Metallic  reflection,  156. 
Michelson,  visibility  with  the  double 
slit,  27. 

interferometer,  33,  48. 

applications  of    the  interferom- 
eter. 69. 

visibility    curves    with    interfe- 
rometer, 82. 

absolute  wave  length,  125. 

action  of    magnetism  on    light, 
180. 

echelon  spectroscope,  180. 
Mirrors,  Fresnel.  30. 
Moigno,     1'Abbe,    optical     theories, 

169. 

Molecular  rotation,  132. 
Muller,  absolute  --vave  length,  124. 
Optical  activity,  130. 
Perot,  distribution  in  sources,  82. 
Polari  scope,  134. 

Polarization     rotation   of  the   plane 
of,  130. 


Polarized  light,   qualitative  experi- 
ments in,  127. 

elliptically,  140. 

reflection  of,  150. 
Prism,  83. 

formation  of  an  image  by,  88. 

thickness  of,  92. 
Pulfrich,  total  reflection,  107. 
Purity  of  spectrum,  94,  111. 
Quarter- wave  plate,  146. 
Quincke,  Fresnel  mirrors,  39. 

diffraction  grating,  108. 
Ray  lei  gh,  limit  of  resolution,  18. 

visibility  curves,  78. 

resolving  power  of  prisms,  91-96. 

plane  gratings,  108,  115. 

Brewster's  law,  151. 
Reflection,  loss  of  phase  on,  66. 

total,  105. 

of  polarized  light,  150. 

metallic,  156. 
Refraction,  index  of,  definition  of,  85. 

determination,    with   the    inter- 
ferometer, 61. 

with  the  prism  spectrometer,  96. 

angle  of,  84. 

determination  of  by  total  reflec- 
tion, 105. 

of  metals,  157. 

relation  to  dielectric   constant, 

173, 
Resolution,  limit  of,  15,  17. 

of  prism,  93. 
Resolving  power,  of  prism,  95. 

determination  of,  102. 
.  of  grating,  111. 

determination  of,  114. 
Rood,  reflection  of  polarized  light, 

152. 
Rotation,  specific,  132. 

naolecular,  132. 
Rowland,  plane  gratings,  108,  115. 

concave  gratings,  116. 

relative  wave  lengths,  125. 
Runge,  concave  gratings,  116. 

regularities  in  spectra,  177. 


196 


Sarasin,  rotary  polarization,  181. 
Schonrock,  rotary   polarization,  133. 
Schwerd,  diffraction,  18. 
Sodium,     determination     of     wave 
lengths,  55. 

ratio    of    wave    lengths   of   the 
lines  DI  and  D2,  59. 

distribution  of  a  single  line,  82. 
Soret,  rotary  polarization,  131. 
Spectrometer,  83. 

adjustments  of,  97. 
Spectrophotometer,  159. 
Spectrum,  purity  of,  94. 

order  of,  110. 


Spectrum,  normal,  111. 

absorption,  164. 

Stefan,  rotary  polarization,  133. 
Thomson,    J.    J.,    electron    theory, 

181. 
Velocity,    of    light,  equals   ratio  of 

units,  172. 

Visibility,  with  the  double  slit,  25. 
with  the  interferometer,  60,  70. 
Wave  length,  absolute  determination 

of,  124. 

Wave  theory,  166. 
Zeeman,    action    of    magnetism    on 

light,  178. 


J 


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